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<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>∫</mo><mi>d</mi><mi>θ</mi><mo>∫</mo><mi>r</mi><mi>d</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">\int d\theta \int rdr</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.805em;"></span><span class="strut bottom" style="height:1.11112em;vertical-align:-0.30612em;"></span><span class="base textstyle uncramped"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="mord mathit">d</span><span class="mord mathit" style="margin-right:0.02778em;">θ</span><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="mord mathit">d</span><span class="mord mathit" style="margin-right:0.02778em;">r</span></span></span></span></p></li> <li><p>函数在开区间连续，如果区间端点的单侧极限存在，则函数有界</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>)</mo><mo>=</mo><msub><mi>max</mi><mrow><mi>a</mi><mo>≤</mo><msub><mi>x</mi><mn>0</mn></msub><mo>≤</mo><mi>b</mi></mrow></msub><mo>{</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>}</mo></mrow><annotation encoding="application/x-tex">f(x_0) = \max_{a \le x_0 \le b} \{f(x)\}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1.005em;vertical-align:-0.255em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mrel">=</span><span class="mop"><span class="mop">max</span><span class="vlist"><span style="top:0.15000000000000002em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">a</span><span class="mrel">≤</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.07142857142857144em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathrm">0</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">≤</span><span class="mord mathit">b</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">{</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mclose">}</span></span></span></span>且 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>)</mo><mo>&gt;</mo><mi>f</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo separator="true">,</mo><mi>f</mi><mo>(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>)</mo><mo>&gt;</mo><mi>f</mi><mo>(</mo><mi>b</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">f(x_0) &gt; f(a), f(x_0) &gt; f(b)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mrel">&gt;</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">a</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mrel">&gt;</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">b</span><span class="mclose">)</span></span></span></span>,f（x）连续，则根据费马定理 有 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>f</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f'(x_0) = 0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.751892em;"></span><span class="strut bottom" style="height:1.001892em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathrm">0</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mo>∬</mo><mi>D</mi></msub><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>)</mo><mi>d</mi><mi>σ</mi><mo>=</mo><msub><mo>∬</mo><mi>D</mi></msub><mi>x</mi><mi>d</mi><mi>σ</mi><mo>+</mo><msub><mo>∬</mo><mi>D</mi></msub><mi>y</mi><mi>d</mi><mi>σ</mi><mo>=</mo><mo>(</mo><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mo>+</mo><mover accent="true"><mi>y</mi><mo>¯</mo></mover><mo>)</mo><mo>⋅</mo><msub><mi>S</mi><mi>D</mi></msub></mrow><annotation encoding="application/x-tex">\iint_D (x + y)d\sigma = \iint_D x d\sigma + \iint_D yd\sigma = (\bar x + \bar y)\cdot S_D</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.805em;"></span><span class="strut bottom" style="height:1.161em;vertical-align:-0.356em;"></span><span class="base textstyle uncramped"><span class="mop"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0004999999999999727em;">∬</span><span class="vlist"><span style="top:0.356em;margin-right:0.05em;margin-left:-0.19445em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit" style="margin-right:0.02778em;">D</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mbin">+</span><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathit">d</span><span class="mord mathit" style="margin-right:0.03588em;">σ</span><span class="mrel">=</span><span class="mop"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0004999999999999727em;">∬</span><span class="vlist"><span style="top:0.356em;margin-right:0.05em;margin-left:-0.19445em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit" style="margin-right:0.02778em;">D</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit">x</span><span class="mord mathit">d</span><span class="mord mathit" style="margin-right:0.03588em;">σ</span><span class="mbin">+</span><span class="mop"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0004999999999999727em;">∬</span><span class="vlist"><span style="top:0.356em;margin-right:0.05em;margin-left:-0.19445em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit" style="margin-right:0.02778em;">D</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="mord mathit">d</span><span class="mord mathit" style="margin-right:0.03588em;">σ</span><span class="mrel">=</span><span class="mopen">(</span><span class="mord accent"><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord mathit">x</span></span><span style="top:0em;margin-left:0.05556em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="accent-body"><span>¯</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord accent"><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord mathit" style="margin-right:0.03588em;">y</span></span><span style="top:0em;margin-left:0.11112em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="accent-body"><span>¯</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mbin">⋅</span><span class="mord"><span class="mord mathit" style="margin-right:0.05764em;">S</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.05764em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit" style="margin-right:0.02778em;">D</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mo separator="true">,</mo><mover accent="true"><mi>y</mi><mo>¯</mo></mover><mo>)</mo></mrow><annotation encoding="application/x-tex">(\bar x, \bar y)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord accent"><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord mathit">x</span></span><span style="top:0em;margin-left:0.05556em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="accent-body"><span>¯</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mpunct">,</span><span class="mord accent"><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord mathit" style="margin-right:0.03588em;">y</span></span><span style="top:0em;margin-left:0.11112em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="accent-body"><span>¯</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span></span></span></span> 是区域D的形心， 对于圆来说就是圆心，通过形心反求积分比较简单</p></li> <li><p><img src="/assets/img/image-20201206201318297.3b604398.png" alt="image-20201206201318297"></p></li> <li><p>证明反常积分 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>∫</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\int f(x)dx</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.805em;"></span><span class="strut bottom" style="height:1.11112em;vertical-align:-0.30612em;"></span><span class="base textstyle uncramped"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mord mathit">d</span><span class="mord mathit">x</span></span></span></span> (包含瑕点或无穷大,如果同时包含则需要拆开讨论)收敛，可以找出f(x)的等价无穷小 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mi>x</mi><mi>p</mi></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{x^p}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.289em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathit">p</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span>, 根据小小取小，大大取大原则，x趋于正无穷大时， p要求大于1；趋于 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mn>0</mn><mo>+</mo></msup></mrow><annotation encoding="application/x-tex">0^+</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.771331em;"></span><span class="strut bottom" style="height:0.771331em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathrm">0</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord">+</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>要求p小于1</p></li> <li><p>如果要证明方程的隐函数y是单调递增（递减）函数，而导函数是关于x，y的函数，不能直接看出导数正负时，考虑在x的取值范围内，y是大于0还是小于0（使用归谬法证明），将方程整理成导数的分子部分=g(x, y)的形式，从而判断导数正负</p></li> <li><p>f(x) - xf’(x)可以看做是f(x)的切线在y轴上的截距</p></li> <li><p>做微分的物理应用是求分力时注意根据方向取正负</p></li> <li><p>判断f(x)的零点个数可以转化为两个图像相交的问题，用画图法观察出零点个数</p></li> <li><p>如果两个二次型的惯性指数相同，则这两个二次型合同</p></li> <li><p><img src="/assets/img/image-20201203114236071.32b334d5.png" alt="image-20201203114236071"></p></li> <li><p>二重积分求极限用二重积分的中值定理，二重积分=<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>ξ</mi><mo separator="true">,</mo><mi>η</mi><mo>)</mo><mo>⋅</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">f(\xi, \eta)\cdot S</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.04601em;">ξ</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.03588em;">η</span><span class="mclose">)</span><span class="mbin">⋅</span><span class="mord mathit" style="margin-right:0.05764em;">S</span></span></span></span>, S为积分区域的面积</p></li> <li><p>求f(x, y)在闭区域内的极值，先求出区域内部的驻点（不需要证明是否是极值点），再使用拉格朗日乘数法求边界上的最值（令各个变量偏导=0）（如果可以消去y，也可以转化为一元函数极值问题），综合得到闭区域的最值</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">\int_0^1 \sqrt{1-x^2}dx = \frac{\pi}{4}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.913389em;"></span><span class="strut bottom" style="height:1.269509em;vertical-align:-0.35612em;"></span><span class="base textstyle uncramped"><span class="mop"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="vlist"><span style="top:0.35612em;margin-left:-0.19445em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span style="top:-0.41900000000000004em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sqrt mord"><span class="sqrt-sign" style="top:-0.07338899999999993em;"><span class="style-wrap reset-textstyle textstyle uncramped">√</span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord textstyle cramped"><span class="mord mathrm">1</span><span class="mbin">−</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.289em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span style="top:-0.8333889999999999em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="mord mathit">d</span><span class="mord mathit">x</span><span class="mrel">=</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">4</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.03588em;">π</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mi>a</mi></mrow><mi>a</mi></msubsup><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\int_{-a}^a f(x)dx = 0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.805em;"></span><span class="strut bottom" style="height:1.219451em;vertical-align:-0.414451em;"></span><span class="base textstyle uncramped"><span class="mop"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="vlist"><span style="top:0.35612em;margin-left:-0.19445em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord">−</span><span class="mord mathit">a</span></span></span></span><span style="top:-0.41900000000000004em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">a</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mord mathit">d</span><span class="mord mathit">x</span><span class="mrel">=</span><span class="mord mathrm">0</span></span></span></span>是 f(x)为奇函数的充要条件（通过对a求导证明）</p></li> <li><p>多元函数不适用洛必达法则和单调有界准则</p></li> <li><p>二阶线性微分方程的右边是关于 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>cos</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\cos x</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mop">cos</span><span class="mord mathit">x</span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>sin</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\sin x</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.66786em;"></span><span class="strut bottom" style="height:0.66786em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mop">sin</span><span class="mord mathit">x</span></span></span></span>的表达式，可以通过观察法求出的特解</p></li> <li><p>与取整函数有关的题目 ，容易用到夹逼准则</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>3</mn></msub><mo>)</mo><mo>=</mo><msup><mi>x</mi><mi>T</mi></msup><mi>A</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">f(x_1, x_2, x_3) = x^TAx</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8413309999999999em;"></span><span class="strut bottom" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mrel">=</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">T</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit">A</span><span class="mord mathit">x</span></span></span></span>为二次型,<em>不</em>代表A就是二次型矩阵，除非A是实对称矩阵</p></li> <li><p>f(x)是周期T的连续函数，关于a的变限积分等式证明，考虑构造函数F(a)并求导</p></li> <li><p>f(x)是周期T的连续函数，则 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>T</mi></msubsup><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\int_0^T f(t)dt</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.897331em;"></span><span class="strut bottom" style="height:1.253451em;vertical-align:-0.35612em;"></span><span class="base textstyle uncramped"><span class="mop"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="vlist"><span style="top:0.35612em;margin-left:-0.19445em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span style="top:-0.41900000000000004em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">T</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">t</span><span class="mclose">)</span><span class="mord mathit">d</span><span class="mord mathit">t</span></span></span></span>的积分区间具有平移不变性</p></li> <li><p><img src="/assets/img/image-20201202145549414.9127f6b8.png" alt="image-20201202145549414"></p></li> <li><p><img src="/assets/img/image-20201202145433384.184a14e7.png" alt="image-20201202145433384"></p></li> <li><p>证明方程的根为实数，只需要求出方程的根的表达式（由实数参数组成）即可</p></li> <li><p>已知二重特征值的两个线性无关的特征向量，可以通过不同特征值的特征向量正交的条件求出第三个特征向量</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mo>∫</mo><mi>a</mi><mi>x</mi></msubsup><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\int_a^x f(t)dt</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.805em;"></span><span class="strut bottom" style="height:1.16112em;vertical-align:-0.35612em;"></span><span class="base textstyle uncramped"><span class="mop"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="vlist"><span style="top:0.35612em;margin-left:-0.19445em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">a</span></span></span><span style="top:-0.41900000000000004em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">x</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">t</span><span class="mclose">)</span><span class="mord mathit">d</span><span class="mord mathit">t</span></span></span></span>具有周期T是f(x)具有周期T的<em>充分非必要条件</em></p></li> <li><p>四个选项：极限不存在、极限存在当不连续、连续但不可导、可导，优先用定义法检验可导</p></li> <li><p>定义法检验在 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x=0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.64444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">x</span><span class="mrel">=</span><span class="mord mathrm">0</span></span></span></span>处是否可导：用 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>f</mi><mo>+</mo><mrow><mi mathvariant="normal">′</mi></mrow></msubsup><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></msub><mfrac><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>f</mi><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mrow><mi>x</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">f'_+(0)= \lim_{x \to 0^+} \frac{f(x) - f(0)}{x}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.01em;"></span><span class="strut bottom" style="height:1.355em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:0.24700000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord">+</span></span></span><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathrm">0</span><span class="mclose">)</span><span class="mrel">=</span><span class="mop"><span class="mop">lim</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span><span class="mrel">→</span><span class="mord"><span class="mord mathrm">0</span><span class="vlist"><span style="top:-0.289em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord">+</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.485em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mbin">−</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathrm">0</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span>与 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>f</mi><mo>−</mo><mrow><mi mathvariant="normal">′</mi></mrow></msubsup><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><msup><mn>0</mn><mo>−</mo></msup></mrow></msub><mfrac><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>f</mi><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mrow><mi>x</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">f'_-(0) = \lim_{x \to 0^-} \frac{f(x)-f(0)}{x}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.01em;"></span><span class="strut bottom" style="height:1.355em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:0.24700000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord">−</span></span></span><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathrm">0</span><span class="mclose">)</span><span class="mrel">=</span><span class="mop"><span class="mop">lim</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span><span class="mrel">→</span><span class="mord"><span class="mord mathrm">0</span><span class="vlist"><span style="top:-0.289em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord">−</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.485em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mbin">−</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathrm">0</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span> 是否相等来证明，</p></li> <li><p>二元函数的极值：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi><mi>C</mi><mo>−</mo><msup><mi>B</mi><mn>2</mn></msup><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">AC-B^2 &gt;0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:0.897438em;vertical-align:-0.08333em;"></span><span class="base textstyle uncramped"><span class="mord mathit">A</span><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="mbin">−</span><span class="mord"><span class="mord mathit" style="margin-right:0.05017em;">B</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">&gt;</span><span class="mord mathrm">0</span></span></span></span>有极值，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">A&gt;0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.72243em;vertical-align:-0.0391em;"></span><span class="base textstyle uncramped"><span class="mord mathit">A</span><span class="mrel">&gt;</span><span class="mord mathrm">0</span></span></span></span>有极小值，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">A&lt;0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.72243em;vertical-align:-0.0391em;"></span><span class="base textstyle uncramped"><span class="mord mathit">A</span><span class="mrel">&lt;</span><span class="mord mathrm">0</span></span></span></span>有极大值；<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi><mi>C</mi><mo>−</mo><msup><mi>B</mi><mn>2</mn></msup><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">AC-B^2 &lt; 0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:0.897438em;vertical-align:-0.08333em;"></span><span class="base textstyle uncramped"><span class="mord mathit">A</span><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="mbin">−</span><span class="mord"><span class="mord mathit" style="margin-right:0.05017em;">B</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">&lt;</span><span class="mord mathrm">0</span></span></span></span>无极值；<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi><mi>C</mi><mo>−</mo><msup><mi>B</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">AC-B^2=0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:0.897438em;vertical-align:-0.08333em;"></span><span class="base textstyle uncramped"><span class="mord mathit">A</span><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="mbin">−</span><span class="mord"><span class="mord mathit" style="margin-right:0.05017em;">B</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord mathrm">0</span></span></span></span>不确定</p></li> <li><p>一般使用伴随矩阵求逆比用初等行变换求逆要快</p></li> <li><p>求极限如果含有变限积分，使用积分中值定理一般比直接洛必达方便</p></li> <li><p>实系数奇数次方程至少有一个根</p></li> <li><p><img src="/assets/img/image-20201201105919346.625badcd.png" alt="image-20201201105919346"></p></li> <li><p>f^{(n)} = 0 至多k个根\Rightarrow f(x)=0至多k+n个根</p></li> <li><p>求极限需要验证左极限是否等于右极限</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><mo>(</mo><mo>−</mo><mn>1</mn><msup><mo>)</mo><mi>n</mi></msup><msup><mi>x</mi><mi>n</mi></msup><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\frac{1}{1+x} = 1 - x + x^2 - x^3+\cdots +(-1)^nx^n,-1 \lt x \lt 1</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.2484389999999999em;vertical-align:-0.403331em;"></span><span class="base textstyle uncramped"><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">1</span><span class="mbin">+</span><span class="mord mathit">x</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mrel">=</span><span class="mord mathrm">1</span><span class="mbin">−</span><span class="mord mathit">x</span><span class="mbin">+</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">−</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="minner">⋯</span><span class="mbin">+</span><span class="mopen">(</span><span class="mord">−</span><span class="mord mathrm">1</span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mpunct">,</span><span class="mord">−</span><span class="mord mathrm">1</span><span class="mrel">&lt;</span><span class="mord mathit">x</span><span class="mrel">&lt;</span><span class="mord mathrm">1</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mo>−</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msup><mi>x</mi><mi>n</mi></msup><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\frac{1}{1-x}=1+x+x^2+\cdots+x^n,-1\lt x\lt1</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.2484389999999999em;vertical-align:-0.403331em;"></span><span class="base textstyle uncramped"><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">1</span><span class="mbin">−</span><span class="mord mathit">x</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mrel">=</span><span class="mord mathrm">1</span><span class="mbin">+</span><span class="mord mathit">x</span><span class="mbin">+</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="minner">⋯</span><span class="mbin">+</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mpunct">,</span><span class="mord">−</span><span class="mord mathrm">1</span><span class="mrel">&lt;</span><span class="mord mathit">x</span><span class="mrel">&lt;</span><span class="mord mathrm">1</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>tan</mi><mi>x</mi><mo>∼</mo><mi>x</mi><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>o</mi><mo>(</mo><msup><mi>x</mi><mn>3</mn></msup><mo>)</mo></mrow><annotation encoding="application/x-tex">\tan x \sim x+\frac{1}{3}x^3+o(x^3)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mop">tan</span><span class="mord mathit">x</span><span class="mrel">∼</span><span class="mord mathit">x</span><span class="mbin">+</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">3</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord mathit">o</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>sin</mi><mi>x</mi><mo>∼</mo><mi>x</mi><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>o</mi><mo>(</mo><msup><mi>x</mi><mn>3</mn></msup><mo>)</mo></mrow><annotation encoding="application/x-tex">\sin x \sim x - \frac{1}{6}x^3+o(x^3)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mop">sin</span><span class="mord mathit">x</span><span class="mrel">∼</span><span class="mord mathit">x</span><span class="mbin">−</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">6</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord mathit">o</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>cos</mi><mi>x</mi><mo>∼</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mn>4</mn></mrow></mfrac><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mi>o</mi><mo>(</mo><msup><mi>x</mi><mn>4</mn></msup><mo>)</mo></mrow><annotation encoding="application/x-tex">\cos x \sim 1-\frac{1}{2}x^2+\frac{1}{24}x^4+o(x^4)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mop">cos</span><span class="mord mathit">x</span><span class="mrel">∼</span><span class="mord mathrm">1</span><span class="mbin">−</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span><span class="mord mathrm">4</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">4</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord mathit">o</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">4</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>arctan</mi><mi>x</mi><mo>∼</mo><mi>x</mi><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>o</mi><mo>(</mo><msup><mi>x</mi><mn>3</mn></msup><mo>)</mo></mrow><annotation encoding="application/x-tex">\arctan x \sim x-\frac{1}{3}x^3+o(x^3)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mop">arctan</span><span class="mord mathit">x</span><span class="mrel">∼</span><span class="mord mathit">x</span><span class="mbin">−</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">3</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord mathit">o</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>arcsin</mi><mi>x</mi><mo>∼</mo><mi>x</mi><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>o</mi><mo>(</mo><msup><mi>x</mi><mn>3</mn></msup><mo>)</mo></mrow><annotation encoding="application/x-tex">\arcsin x \sim x+\frac{1}{6}x^3+o(x^3)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mop">arcsin</span><span class="mord mathit">x</span><span class="mrel">∼</span><span class="mord mathit">x</span><span class="mbin">+</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">6</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord mathit">o</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo><mo>∼</mo><mi>x</mi><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>o</mi><mo>(</mo><msup><mi>x</mi><mn>3</mn></msup><mo>)</mo></mrow><annotation encoding="application/x-tex">\ln (1+x) \sim x -\frac{1}{2}x^2+\frac{1}{3}x^3+o(x^3)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mop">ln</span><span class="mopen">(</span><span class="mord mathrm">1</span><span class="mbin">+</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mrel">∼</span><span class="mord mathit">x</span><span class="mbin">−</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">3</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord mathit">o</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>e</mi><mi>x</mi></msup><mo>=</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>o</mi><mo>(</mo><msup><mi>x</mi><mn>3</mn></msup><mo>)</mo></mrow><annotation encoding="application/x-tex">e^x=1+x+\frac{1}{2}x^2+\frac{1}{6}x^3+o(x^3)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">e</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">x</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord mathrm">1</span><span class="mbin">+</span><span class="mord mathit">x</span><span class="mbin">+</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">6</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord mathit">o</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><msup><mo>)</mo><mi>α</mi></msup><mo>=</mo><mn>1</mn><mo>+</mo><mi>α</mi><mi>x</mi><mo>+</mo><mfrac><mrow><mi>α</mi><mo>(</mo><mi>α</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mi>α</mi><mo>(</mo><mi>α</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>α</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow><mrow><mn>6</mn></mrow></mfrac><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>o</mi><mo>(</mo><msup><mi>x</mi><mn>3</mn></msup><mo>)</mo></mrow><annotation encoding="application/x-tex">(1+x)^\alpha = 1+\alpha x+\frac{\alpha (\alpha -1)}{2}x^2+\frac{\alpha (\alpha-1)(\alpha-2)}{6}+x^3+o(x^3)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.01em;"></span><span class="strut bottom" style="height:1.355em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathrm">1</span><span class="mbin">+</span><span class="mord mathit">x</span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.0037em;">α</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord mathrm">1</span><span class="mbin">+</span><span class="mord mathit" style="margin-right:0.0037em;">α</span><span class="mord mathit">x</span><span class="mbin">+</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.485em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.0037em;">α</span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.0037em;">α</span><span class="mbin">−</span><span class="mord mathrm">1</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">6</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.485em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.0037em;">α</span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.0037em;">α</span><span class="mbin">−</span><span class="mord mathrm">1</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.0037em;">α</span><span class="mbin">−</span><span class="mord mathrm">2</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mbin">+</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord mathit">o</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span></span></span></span>，主要在 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.0037em;">α</span></span></span></span>为分式的时候使用</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>a</mi><mi>n</mi></msup><mo>−</mo><msup><mi>b</mi><mi>b</mi></msup><mo>=</mo><mo>(</mo><mi>a</mi><mo>−</mo><mi>b</mi><mo>)</mo><mo>(</mo><msup><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>b</mi><mo>+</mo><mo>⋯</mo><mo>+</mo><mi>a</mi><msup><mi>b</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><msup><mi>b</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><annotation encoding="application/x-tex">a^n-b^b = (a-b)(a^{n-1}+a^{n-2}b+\cdots+ab^{n-2}+b^{n-1})</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.849108em;"></span><span class="strut bottom" style="height:1.099108em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">−</span><span class="mord"><span class="mord mathit">b</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">b</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mopen">(</span><span class="mord mathit">a</span><span class="mbin">−</span><span class="mord mathit">b</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">n</span><span class="mbin">−</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">n</span><span class="mbin">−</span><span class="mord mathrm">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit">b</span><span class="mbin">+</span><span class="minner">⋯</span><span class="mbin">+</span><span class="mord mathit">a</span><span class="mord"><span class="mord mathit">b</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">n</span><span class="mbin">−</span><span class="mord mathrm">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord"><span class="mord mathit">b</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">n</span><span class="mbin">−</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>a</mi><mi>n</mi></msup><mo>+</mo><msup><mi>b</mi><mi>b</mi></msup><mo>=</mo><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>)</mo><mo>(</mo><msup><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><msup><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>b</mi><mo>+</mo><mo>⋯</mo><mo>)</mo></mrow><annotation encoding="application/x-tex">a^n+b^b = (a+b)(a^{n-1}-a^{n-2}b+\cdots )</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.849108em;"></span><span class="strut bottom" style="height:1.099108em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord"><span class="mord mathit">b</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">b</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mopen">(</span><span class="mord mathit">a</span><span class="mbin">+</span><span class="mord mathit">b</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">n</span><span class="mbin">−</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">−</span><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">n</span><span class="mbin">−</span><span class="mord mathrm">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit">b</span><span class="mbin">+</span><span class="minner">⋯</span><span class="mclose">)</span></span></span></span></p></li> <li><p>华里士公式：</p>
\int_0^\frac{\pi}{2}sin^nx = \int_0^\frac{\pi}{2}cos^n x = \begin{cases}
\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdot \frac{2}{3}&amp;,n为奇数\\
\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdot \frac{1}{2}\cdot.\frac{\pi}{2}&amp;,n为偶数
\end{cases}

</li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>sinh</mi><mi>x</mi><msup><mo>)</mo><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>=</mo><mi>cosh</mi><mi>x</mi><mo>=</mo><mfrac><mrow><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>x</mi></mrow></msup></mrow><mrow><mn>2</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">(\sinh x)'=\cosh x= \frac{e^x+e^{-x}}{2}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.939765em;"></span><span class="strut bottom" style="height:1.284765em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mop">sinh</span><span class="mord mathit">x</span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mop">cosh</span><span class="mord mathit">x</span><span class="mrel">=</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord"><span class="mord mathit">e</span><span class="vlist"><span style="top:-0.363em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord mathit">x</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord"><span class="mord mathit">e</span><span class="vlist"><span style="top:-0.363em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord scriptscriptstyle uncramped"><span class="mord">−</span><span class="mord mathit">x</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>cosh</mi><mi>x</mi><msup><mo>)</mo><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>=</mo><mi>sinh</mi><mi>x</mi><mo>=</mo><mfrac><mrow><msup><mi>e</mi><mi>x</mi></msup><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>x</mi></mrow></msup></mrow><mrow><mn>2</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">(\cosh x)' = \sinh x = \frac{e^x-e^{-x}}{2}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.939765em;"></span><span class="strut bottom" style="height:1.284765em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mop">cosh</span><span class="mord mathit">x</span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mop">sinh</span><span class="mord mathit">x</span><span class="mrel">=</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord"><span class="mord mathit">e</span><span class="vlist"><span style="top:-0.363em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord mathit">x</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">−</span><span class="mord"><span class="mord mathit">e</span><span class="vlist"><span style="top:-0.363em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord scriptscriptstyle uncramped"><span class="mord">−</span><span class="mord mathit">x</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>sinh</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\sinh x</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mop">sinh</span><span class="mord mathit">x</span></span></span></span>的反函数是 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ln</mi><mo>(</mo><mi>x</mi><mo>+</mo><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>)</mo></mrow></msqrt></mrow><annotation encoding="application/x-tex">\ln (x+\sqrt{x^2+1)}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.9350050000000001em;"></span><span class="strut bottom" style="height:1.24001em;vertical-align:-0.3050049999999999em;"></span><span class="base textstyle uncramped"><span class="mop">ln</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mbin">+</span><span class="sqrt mord"><span class="sqrt-sign" style="top:-0.04500500000000007em;"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing size1">√</span></span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord textstyle cramped"><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.289em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord mathrm">1</span><span class="mclose">)</span></span></span><span style="top:-0.855005em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span></span></span></span></p></li> <li><p>\cosh x ,x\in [1,+\infin]反函数是 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>ln</mi><mo>(</mo><mi>x</mi><mo>+</mo><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></msqrt><mo>)</mo></mrow><annotation encoding="application/x-tex">\ln (x + \sqrt{x^2-1})</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.913389em;"></span><span class="strut bottom" style="height:1.163389em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mop">ln</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mbin">+</span><span class="sqrt mord"><span class="sqrt-sign" style="top:-0.07338899999999993em;"><span class="style-wrap reset-textstyle textstyle uncramped">√</span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord textstyle cramped"><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.289em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">−</span><span class="mord mathrm">1</span></span></span><span style="top:-0.8333889999999999em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="mclose">)</span></span></span></span></p></li> <li><p>(\textrm{arcsinh}\space x)' = \frac{1}{\sqrt{1+x^2}}</p></li> <li><p>参数方程的弧长是 s = \int \sqrt{x_t'^2+y_t'^2} dt ,直角坐标系的弧长是 $s=\int \sqrt{1+y'}dx $ 极坐标系的求弧长 s = \int\sqrt{r'^2+r^2}d\theta</p></li> <li><p>(\textrm{arccosh}\space x)' = \frac{1}{\sqrt{x^2-1}}</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>arcsin</mi><mi>x</mi><msup><mo>)</mo><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></mrow></mfrac></mrow><annotation encoding="application/x-tex">(\arcsin x)' = \frac{1}{\sqrt{1-x^2}}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.6951080000000003em;vertical-align:-0.8500000000000002em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mop">arcsin</span><span class="mord mathit">x</span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.6280945000000002em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="sqrt mord"><span class="sqrt-sign" style="top:0.031293571428571365em;"><span class="style-wrap reset-scriptstyle textstyle uncramped">√</span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1.4285714285714286em;">​</span></span><span class="mord scriptstyle cramped"><span class="mord mathrm">1</span><span class="mbin">−</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.289em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span style="top:-1.0544207142857145em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1.4285714285714286em;">​</span></span><span class="reset-scriptstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1.4285714285714286em;">​</span></span>​</span></span></span></span></span></span><span style="top:-0.2300000000000001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>arccos</mi><mi>x</mi><msup><mo>)</mo><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>=</mo><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></mrow></mfrac></mrow><annotation encoding="application/x-tex">(\arccos x)' = -\frac{1}{\sqrt{1-x^2}}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.6951080000000003em;vertical-align:-0.8500000000000002em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mop">arccos</span><span class="mord mathit">x</span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord">−</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.6280945000000002em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="sqrt mord"><span class="sqrt-sign" style="top:0.031293571428571365em;"><span class="style-wrap reset-scriptstyle textstyle uncramped">√</span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1.4285714285714286em;">​</span></span><span class="mord scriptstyle cramped"><span class="mord mathrm">1</span><span class="mbin">−</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.289em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span style="top:-1.0544207142857145em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1.4285714285714286em;">​</span></span><span class="reset-scriptstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1.4285714285714286em;">​</span></span>​</span></span></span></span></span></span><span style="top:-0.2300000000000001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>arctan</mi><mi>x</mi><msup><mo>)</mo><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">(\arctan x)' = \frac{1}{1+x^2}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.2484389999999999em;vertical-align:-0.403331em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mop">arctan</span><span class="mord mathit">x</span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">1</span><span class="mbin">+</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.289em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>3</mn><mn>2</mn></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mi>n</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>(</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>6</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">1+2^2+3^2+\cdots+n^2 = \frac{n(n+1)(2n+1)}{6}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.01em;"></span><span class="strut bottom" style="height:1.355em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">1</span><span class="mbin">+</span><span class="mord"><span class="mord mathrm">2</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord"><span class="mord mathrm">3</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="minner">⋯</span><span class="mbin">+</span><span class="mord"><span class="mord mathit">n</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">6</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.485em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">n</span><span class="mopen">(</span><span class="mord mathit">n</span><span class="mbin">+</span><span class="mord mathrm">1</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathrm">2</span><span class="mord mathit">n</span><span class="mbin">+</span><span class="mord mathrm">1</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>sin</mi><mn>3</mn><mi>α</mi><mo>=</mo><mo>−</mo><mn>4</mn><msup><mi>sin</mi><mn>3</mn></msup><mi>α</mi><mo>+</mo><mn>3</mn><mi>sin</mi><mi>α</mi></mrow><annotation encoding="application/x-tex">\sin 3\alpha = -4\sin^3 \alpha + 3\sin \alpha</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:0.897438em;vertical-align:-0.08333em;"></span><span class="base textstyle uncramped"><span class="mop">sin</span><span class="mord mathrm">3</span><span class="mord mathit" style="margin-right:0.0037em;">α</span><span class="mrel">=</span><span class="mord">−</span><span class="mord mathrm">4</span><span class="mop"><span class="mop">sin</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.0037em;">α</span><span class="mbin">+</span><span class="mord mathrm">3</span><span class="mop">sin</span><span class="mord mathit" style="margin-right:0.0037em;">α</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>cos</mi><mn>3</mn><mi>α</mi><mo>=</mo><mn>4</mn><msup><mi>cos</mi><mn>3</mn></msup><mi>α</mi><mo>−</mo><mn>3</mn><mi>cos</mi><mi>α</mi></mrow><annotation encoding="application/x-tex">\cos 3\alpha = 4\cos^3 \alpha - 3\cos \alpha</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:0.897438em;vertical-align:-0.08333em;"></span><span class="base textstyle uncramped"><span class="mop">cos</span><span class="mord mathrm">3</span><span class="mord mathit" style="margin-right:0.0037em;">α</span><span class="mrel">=</span><span class="mord mathrm">4</span><span class="mop"><span class="mop">cos</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.0037em;">α</span><span class="mbin">−</span><span class="mord mathrm">3</span><span class="mop">cos</span><span class="mord mathit" style="margin-right:0.0037em;">α</span></span></span></span></p></li> <li><p>(abc)’ = a’bc+ab’c+abc</p></li> <li><p>有限个无穷小相加还是无穷小</p></li> <li><p>夹逼准则的特殊形式 |f(x)| <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\le</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.63597em;"></span><span class="strut bottom" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mrel">≤</span></span></span></span> g(x), <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mi>α</mi></mrow></msub><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\lim_{x \to \alpha} g(x) = a</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mop"><span class="mop">lim</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span><span class="mrel">→</span><span class="mord mathit" style="margin-right:0.0037em;">α</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathit">a</span></span></span></span> ， 则 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mi>α</mi></mrow></msub><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\lim_{x \to \alpha} f(x) = a</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mop"><span class="mop">lim</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span><span class="mrel">→</span><span class="mord mathit" style="margin-right:0.0037em;">α</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathit">a</span></span></span></span></p></li> <li><p>向量组 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>α</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_i</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.0037em;">α</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.0037em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">i</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>线性无关， 则向量组 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>α</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_i</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.0037em;">α</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.0037em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">i</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>和向量组 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>β</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\beta_i</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.05278em;">β</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.05278em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">i</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>的等价是 向量组 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>β</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\beta_i</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.05278em;">β</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.05278em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">i</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>线性无关的充分非必要条件</p></li> <li><p>一定要先化简再求导</p></li> <li><p>有界乘无穷小还是无穷小</p></li> <li><p>如果题目只给出在点 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>上可导，没有说区间内可导，这不能用拉格朗日定理，只能用带佩亚诺余项的泰勒公式</p></li> <li><p><img src="/assets/img/image-20201120103109511.d53fecaf.png" alt="image-20201120103109511"></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>lim</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>±</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>=</mo><mi>lim</mi><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>±</mo><mi>lim</mi><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\lim (f(x) \pm g(x)) = \lim f(x)\pm\lim g(x)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mop">lim</span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mbin">±</span><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mclose">)</span><span class="mrel">=</span><span class="mop">lim</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mbin">±</span><span class="mop">lim</span><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>lim</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>×</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>=</mo><mi>lim</mi><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>×</mo><mi>lim</mi><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\lim (f(x) \times g(x)) = \lim f(x)\times\lim g(x)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mop">lim</span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mbin">×</span><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mclose">)</span><span class="mrel">=</span><span class="mop">lim</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mbin">×</span><span class="mop">lim</span><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>lim</mi><mi>f</mi><mo>[</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>]</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>lim</mi><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></mrow><annotation encoding="application/x-tex">\lim f[g(x)]=f(\lim g(x))</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mop">lim</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">[</span><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mclose">]</span><span class="mrel">=</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mop">lim</span><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mclose">)</span></span></span></span>，前提是 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>lim</mi><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\lim g(x)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mop">lim</span><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span>必须存在</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>P</mi><mi>T</mi></msup><mi>P</mi></mrow><annotation encoding="application/x-tex">P^TP</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8413309999999999em;"></span><span class="strut bottom" style="height:0.8413309999999999em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">P</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">T</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.13889em;">P</span></span></span></span>是P的列向量间的内积，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><msup><mi>P</mi><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">PP^T</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8413309999999999em;"></span><span class="strut bottom" style="height:0.8413309999999999em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">P</span><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">P</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">T</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>是P的行向量间的内积</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>lim</mi><mfrac><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>lim</mi><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>lim</mi><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\lim \frac{f(x)}{g(x)}=\frac{\lim f(x)}{\lim g(x)}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.01em;"></span><span class="strut bottom" style="height:1.53em;vertical-align:-0.52em;"></span><span class="base textstyle uncramped"><span class="mop">lim</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.34500000000000003em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.485em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mrel">=</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.34500000000000003em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mop">lim</span><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.485em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mop">lim</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span>前提是 \lim g(x)和 \lim f(x)存在且不为0</p></li> <li><p>不定积分可以写成变限积分的形式，不定积分出现在不等式证明的时候</p></li> <li><p>二元函数可微的必要条件：连续；偏导数存在</p></li> <li><p>二元函数可微的充分条件：偏导数连续</p></li> <li><p>二元函数在点(0,0)可微的充要条件: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>lim</mi><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>→</mo><mn>0</mn><mo separator="true">,</mo><mi mathvariant="normal">Δ</mi><mi>y</mi><mo>→</mo><mn>0</mn></mrow></msub><mfrac><mrow><mi>f</mi><mo>(</mo><mi mathvariant="normal">Δ</mi><mi>x</mi><mo separator="true">,</mo><mi mathvariant="normal">Δ</mi><mi>y</mi><mo>)</mo><mo>−</mo><mi>f</mi><mo>(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo>)</mo><mo>−</mo><msubsup><mi>f</mi><mi>x</mi><mrow><mi mathvariant="normal">′</mi></mrow></msubsup><mrow><mn>0</mn><mo separator="true">,</mo><mn>0</mn></mrow><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>−</mo><msubsup><mi>f</mi><mi>y</mi><mrow><mi mathvariant="normal">′</mi></mrow></msubsup><mrow><mn>0</mn><mo separator="true">,</mo><mn>0</mn></mrow><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow><mrow><msqrt><mrow><mi mathvariant="normal">Δ</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi mathvariant="normal">Δ</mi><msup><mi>y</mi><mn>2</mn></msup></mrow></msqrt></mrow></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\lim_ {\Delta x \to 0, \Delta y \to 0} \frac{f(\Delta x , \Delta y) -  f(0,0) - f'_x{0, 0}\Delta x - f'_y{0, 0}\Delta y}{\sqrt{\Delta x^2 + \Delta y^2}} = 0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.112em;"></span><span class="strut bottom" style="height:1.9620000000000004em;vertical-align:-0.8500000000000003em;"></span><span class="base textstyle uncramped"><span class="mop"><span class="mop">lim</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">Δ</span><span class="mord mathit">x</span><span class="mrel">→</span><span class="mord mathrm">0</span><span class="mpunct">,</span><span class="mord mathrm">Δ</span><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="mrel">→</span><span class="mord mathrm">0</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.5892060000000001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="sqrt mord"><span class="sqrt-sign" style="top:0.08684857142857161em;"><span class="style-wrap reset-scriptstyle textstyle uncramped">√</span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1.4285714285714286em;">​</span></span><span class="mord scriptstyle cramped"><span class="mord mathrm">Δ</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.289em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord mathrm">Δ</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="vlist"><span style="top:-0.289em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span style="top:-0.9988657142857145em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1.4285714285714286em;">​</span></span><span class="reset-scriptstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1.4285714285714286em;">​</span></span>​</span></span></span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.58012em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathrm">Δ</span><span class="mord mathit">x</span><span class="mpunct">,</span><span class="mord mathrm">Δ</span><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mbin">−</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathrm">0</span><span class="mpunct">,</span><span class="mord mathrm">0</span><span class="mclose">)</span><span class="mbin">−</span><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:0.247em;margin-left:-0.10764em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathit">x</span></span></span><span style="top:-0.363em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord scriptscriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord scriptstyle uncramped"><span class="mord mathrm">0</span><span class="mpunct">,</span><span class="mord mathrm">0</span></span><span class="mord mathrm">Δ</span><span class="mord mathit">x</span><span class="mbin">−</span><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:0.247em;margin-left:-0.10764em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathit" style="margin-right:0.03588em;">y</span></span></span><span style="top:-0.363em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord scriptscriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord scriptstyle uncramped"><span class="mord mathrm">0</span><span class="mpunct">,</span><span class="mord mathrm">0</span></span><span class="mord mathrm">Δ</span><span class="mord mathit" style="margin-right:0.03588em;">y</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mrel">=</span><span class="mord mathrm">0</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>→</mo><mn>0</mn><mo separator="true">,</mo><mi>o</mi><mo>(</mo><msup><mi>x</mi><mi>a</mi></msup><mo>)</mo><mo>+</mo><mi>o</mi><mo>(</mo><msup><mi>x</mi><mi>b</mi></msup><mo>)</mo><mo>=</mo><mi>o</mi><mo>(</mo><msup><mi>x</mi><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mo>(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo>)</mo></mrow></msup><mo>)</mo></mrow><annotation encoding="application/x-tex">x\to 0, o(x^a)+o(x^b)=o(x^{min(a,b)})</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8879999999999999em;"></span><span class="strut bottom" style="height:1.138em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit">x</span><span class="mrel">→</span><span class="mord mathrm">0</span><span class="mpunct">,</span><span class="mord mathit">o</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">a</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mbin">+</span><span class="mord mathit">o</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">b</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathit">o</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">m</span><span class="mord mathit">i</span><span class="mord mathit">n</span><span class="mopen">(</span><span class="mord mathit">a</span><span class="mpunct">,</span><span class="mord mathit">b</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span></span></span></span>,求多项式是多少阶无穷小只看最低非0次幂</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>x</mi><mi>a</mi></msup><mi>o</mi><mo>(</mo><msup><mi>x</mi><mi>b</mi></msup><mo>)</mo><mo>=</mo><mi>o</mi><mo>(</mo><msup><mi>x</mi><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></msup><mo>)</mo></mrow><annotation encoding="application/x-tex">x^ao(x^b)=o(x^{a+b})</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.849108em;"></span><span class="strut bottom" style="height:1.099108em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">a</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit">o</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">b</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathit">o</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">a</span><span class="mbin">+</span><span class="mord mathit">b</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span></span></span></span></p></li> <li><p>x\to \infin, (1+\frac{a}{x})^{bx+d}=e^{ab}</p></li> <li><p>求极限时化简先行（如果某部分极限不为0，可以先把该部分化简为常数），优先找有界函数（一般是三角函数）与无穷小的积；涉及根号式，考虑分子分母同时除于x的某次幂</p></li> <li><p>泰勒不行立即考虑洛必达</p></li> <li><p>如果对dx积分很难求，尝试转换坐标轴，对dy积分</p></li> <li><p>求定积分的分母出现了 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">1+x^2</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:0.897438em;vertical-align:-0.08333em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">1</span><span class="mbin">+</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>的因式，考虑<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>=</mo><mi>tan</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">x=\tan t</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.61508em;"></span><span class="strut bottom" style="height:0.61508em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">x</span><span class="mrel">=</span><span class="mop">tan</span><span class="mord mathit">t</span></span></span></span>,如果出现了 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x^2 - 1</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:0.897438em;vertical-align:-0.08333em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">−</span><span class="mord mathrm">1</span></span></span></span>,考虑 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>=</mo><mi>sec</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">x = \sec t</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.61508em;"></span><span class="strut bottom" style="height:0.61508em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">x</span><span class="mrel">=</span><span class="mop">sec</span><span class="mord mathit">t</span></span></span></span></p></li> <li><p>极坐标封闭区域的面积公式 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msubsup><mo>∫</mo><mi>α</mi><mi>β</mi></msubsup><msup><mi>r</mi><mn>2</mn></msup><mi>d</mi><mi>θ</mi></mrow><annotation encoding="application/x-tex">\frac{1}{2}\int_\alpha^\beta r^2d\theta</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.905108em;"></span><span class="strut bottom" style="height:1.261228em;vertical-align:-0.35612em;"></span><span class="base textstyle uncramped"><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mop"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="vlist"><span style="top:0.35612em;margin-left:-0.19445em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit" style="margin-right:0.0037em;">α</span></span></span><span style="top:-0.41900000000000004em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.05278em;">β</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit">d</span><span class="mord mathit" style="margin-right:0.02778em;">θ</span></span></span></span></p></li> <li><p>对分式全是三角函数的，考虑令 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>=</mo><mi>π</mi><mo>−</mo><mi>u</mi></mrow><annotation encoding="application/x-tex">x = \pi - u</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.58333em;"></span><span class="strut bottom" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="base textstyle uncramped"><span class="mord mathit">x</span><span class="mrel">=</span><span class="mord mathit" style="margin-right:0.03588em;">π</span><span class="mbin">−</span><span class="mord mathit">u</span></span></span></span></p></li> <li><p>某些包含三角函数或指数函数的分式的定积分，可能在使用分部积分法后，消去部分积分项</p></li> <li><p>求极限可以使用拉格朗日中值定理</p></li> <li><p>证明 \int_0^{+\infin} f(x)dx收敛 一般拆成 I_1+I_2 = \int_0^1f(x)dx + \int_1^{+\infin} f(x)dx证明分别都收敛，证明在 x \to 0,f(x)\sim g(x),\int_0^{+\infin} g(x)dx收敛</p></li> <li><p>遇到 x\to \infin的，后面可以考虑换元令 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>x</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">t = \frac{1}{x}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord mathit">t</span><span class="mrel">=</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span></p></li> <li><p>求x的无穷小的阶数直接用泰勒公式，不要用洛必达，泰勒展开式展开到原式结果x的最高次幂不为0为止</p></li> <li><p>根号式也可以用泰勒公式展开</p></li> <li><p>积分与某个参数t无关，说明此积分对t求导结果为0</p></li> <li><p>实对称矩阵乘实对称矩阵，结果不一定是实对称矩阵</p></li> <li><p>矩阵A是正定矩阵当且仅当A的特征值全部大于0</p></li> <li><p>隐函数存在定理：如果 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>F</mi><mi>x</mi><mrow><mi mathvariant="normal">′</mi></mrow></msubsup><mo>(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>0</mn></msub><mo>)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">F'_x(x_0,y_0) = 0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.751892em;"></span><span class="strut bottom" style="height:1.001892em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="vlist"><span style="top:0.247em;margin-left:-0.13889em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">x</span></span></span><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathrm">0</span></span></span></span>，则<em>不</em>能在 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>0</mn></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">(x_0, y_0)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span></span></span></span>的邻域内确定一个隐函数 x=x(y)</p></li> <li><p>证明导数为0，可用费马定理，先找到极值点（函数在区间内可导，在某点的值在邻域内最大或最小）</p></li> <li><p><img src="/assets/img/image-20201130163237449.8428c25f.png" alt="image-20201130163237449">这个二阶导数的分段函数<em>取不到</em>分段点，因为二阶导数分段点左右极限不等</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><mo>+</mo><msup><mi>cos</mi><mi>α</mi></msup><mi>x</mi><mo>∼</mo><mfrac><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">1+\cos^\alpha x \sim \frac{\alpha}{2}x^2</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:1.1591079999999998em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">1</span><span class="mbin">+</span><span class="mop"><span class="mop">cos</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.0037em;">α</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit">x</span><span class="mrel">∼</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.0037em;">α</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></p></li> <li><p>开区间求最值，比较端点的函数极限、不可导点与驻点的函数值的大小</p></li> <li><p><img src="/assets/img/image-20201125161844808.5e8ebb06.png" alt="image-20201125161844808"></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi><msup><mi>A</mi><mi>T</mi></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">AA^T=0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8413309999999999em;"></span><span class="strut bottom" style="height:0.8413309999999999em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">A</span><span class="mord"><span class="mord mathit">A</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">T</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord mathrm">0</span></span></span></span>的充要条件是A = O</p></li> <li><p>\lim_{x \to \infin} x^\frac{1}{x} = 1</p></li> <li><p>\lim _ {n \to \infin} (a_1^n + \cdots + a_m^n) ^ {\frac{1}{n}} = \max\{a_1, \cdots, a_m\}, a_i \ge0</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>r</mi><mo>(</mo><msub><mi>A</mi><mrow><mi>n</mi><mo>×</mo><mi>k</mi></mrow></msub><mo>)</mo><mo>+</mo><mi>r</mi><mo>(</mo><msub><mi>B</mi><mrow><mi>k</mi><mo>×</mo><mi>m</mi></mrow></msub><mo>)</mo><mo>≤</mo><mi>r</mi><mo>(</mo><mi>A</mi><mi>B</mi><mo>)</mo><mo>+</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">r(A_{n\times k}) + r(B_{k\times m}) \le r(AB)+k</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">A</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">n</span><span class="mbin">×</span><span class="mord mathit" style="margin-right:0.03148em;">k</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mbin">+</span><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right:0.05017em;">B</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.05017em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mbin">×</span><span class="mord mathit">m</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mrel">≤</span><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="mopen">(</span><span class="mord mathit">A</span><span class="mord mathit" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mbin">+</span><span class="mord mathit" style="margin-right:0.03148em;">k</span></span></span></span></p></li> <li><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>r</mi><mo>(</mo><msup><mi>A</mi><mo>∗</mo></msup><mo>)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo separator="true">,</mo></mrow></mtd><mtd><mrow><mi>r</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mi>n</mi></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn><mo separator="true">,</mo></mrow></mtd><mtd><mrow><mi>r</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn><mo separator="true">,</mo></mrow></mtd><mtd><mrow><mi>r</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">r(A^*) = \begin{cases}n,&amp; r(A) = n\\
1,&amp; r(A) = n-1\\
0,&amp; r(A) = 0\end{cases}
</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:2.41em;"></span><span class="strut bottom" style="height:4.32em;vertical-align:-1.9099999999999997em;"></span><span class="base displaystyle textstyle uncramped"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">A</span><span class="vlist"><span style="top:-0.413em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord">∗</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mrel">=</span><span class="minner displaystyle textstyle uncramped"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing mult"><span class="vlist"><span style="top:0.9500099999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:0.9500099999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-0.000010000000000287557em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-1.1500100000000002em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-1.4500200000000003em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist"><span style="top:-1.4020000000000001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord mathit">n</span><span class="mpunct">,</span></span></span><span style="top:0.0379999999999997em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord mathrm">1</span><span class="mpunct">,</span></span></span><span style="top:1.4779999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord mathrm">0</span><span class="mpunct">,</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist"><span style="top:-1.4020000000000001em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="mopen">(</span><span class="mord mathit">A</span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathit">n</span></span></span><span style="top:0.0379999999999997em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="mopen">(</span><span class="mord mathit">A</span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathit">n</span><span class="mbin">−</span><span class="mord mathrm">1</span></span></span><span style="top:1.4779999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="mord displaystyle textstyle uncramped"><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="mopen">(</span><span class="mord mathit">A</span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathrm">0</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span></span></p></li> <li><p>符号正负交替的通项，首项含负号用 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mo>−</mo><mn>1</mn><msup><mo>)</mo><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">(-1)^n</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord">−</span><span class="mord mathrm">1</span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>，不含负号用 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mo>−</mo><mn>1</mn><msup><mo>)</mo><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">(-1)^{n+1}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord">−</span><span class="mord mathrm">1</span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">n</span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></p></li> <li><p>对分母可用等价无穷小替换，注意必须是等价而不是高阶，不建议对分母泰勒展开成两项以上</p></li> <li><p>对于 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mn>0</mn><mo>+</mo><mn>0</mn></mrow><mrow><mn>0</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{0+0}{0}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">0</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">0</span><span class="mbin">+</span><span class="mord mathrm">0</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span>型，可以拆成 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mn>0</mn></mrow><mrow><mn>0</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>0</mn></mrow><mrow><mn>0</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{0}{0}+\frac{0}{0}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">0</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">0</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mbin">+</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">0</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">0</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span>简化计算，前提是子部分的极限必须存在</p></li> <li><p>如果使用待定系数法拆有理分式无解，可能是两个多项式之间存在公因式，应消除公因式才能使用待定系数法</p></li> <li><p>如果对一个式子求极限，感觉式子还是很复杂，无从下手，可能原因有两个</p> <ol><li>有界乘无穷小的部分没有被消去</li> <li>公因式的极限已经可求解却还没有换算成常数</li></ol></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>→</mo><mn>0</mn><mo separator="true">,</mo><mi>x</mi><mi>ln</mi><mi>x</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x \to 0, x\ln x \to 0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit">x</span><span class="mrel">→</span><span class="mord mathrm">0</span><span class="mpunct">,</span><span class="mord mathit">x</span><span class="mop">ln</span><span class="mord mathit">x</span><span class="mrel">→</span><span class="mord mathrm">0</span></span></span></span></p></li> <li><p>基本不等式： <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>min</mi><mo>{</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo>}</mo><mo>≤</mo><mfrac><mrow><mn>2</mn></mrow><mrow><msup><mi>a</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mi>b</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>≤</mo><msqrt><mrow><mi>a</mi><mi>b</mi></mrow></msqrt><mo>≤</mo><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>≤</mo><msqrt><mrow><mfrac><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msqrt></mrow><annotation encoding="application/x-tex">\min \{a, b\} \le \frac{2}{a^{-1}+b^{-1}} \le \sqrt{ab} \le \frac{a+b}{2} \le \sqrt{\frac{a^2+b^2}{2}}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.27177em;"></span><span class="strut bottom" style="height:1.84002em;vertical-align:-0.5682499999999999em;"></span><span class="base textstyle uncramped"><span class="mop">min</span><span class="mopen">{</span><span class="mord mathit">a</span><span class="mpunct">,</span><span class="mord mathit">b</span><span class="mclose">}</span><span class="mrel">≤</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:-0.289em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord scriptscriptstyle cramped"><span class="mord">−</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord"><span class="mord mathit">b</span><span class="vlist"><span style="top:-0.289em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord scriptscriptstyle cramped"><span class="mord">−</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mrel">≤</span><span class="sqrt mord"><span class="sqrt-sign" style="top:-0.09221999999999997em;"><span class="style-wrap reset-textstyle textstyle uncramped">√</span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord textstyle cramped"><span class="mord mathit">a</span><span class="mord mathit">b</span></span></span><span style="top:-0.85222em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span><span class="mrel">≤</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">a</span><span class="mbin">+</span><span class="mord mathit">b</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mrel">≤</span><span class="sqrt mord"><span class="sqrt-sign" style="top:-0.08177000000000012em;"><span class="style-wrap reset-textstyle textstyle uncramped"><span class="delimsizing size2">√</span></span></span><span class="vlist"><span style="top:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="mord textstyle cramped"><span class="mord reset-textstyle textstyle cramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:-0.289em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord"><span class="mord mathit">b</span><span class="vlist"><span style="top:-0.289em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span><span style="top:-1.19177em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span><span class="reset-textstyle textstyle uncramped sqrt-line"></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:1em;">​</span></span>​</span></span></span></span></span></span></p></li> <li><p>讨论分段函数的间断点，除了令式子无意义的点，不要忘了分段函数的分割点</p></li> <li><p>极坐标的二重积分的计算公式 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>∬</mo><mi>f</mi><mo>(</mo><mi>ρ</mi><mi>cos</mi><mi>θ</mi><mo separator="true">,</mo><mi>ρ</mi><mi>sin</mi><mi>θ</mi><mo>)</mo><mi>ρ</mi><mi>d</mi><mi>ρ</mi><mi>d</mi><mi>θ</mi></mrow><annotation encoding="application/x-tex">\iint f(\rho\cos \theta, \rho \sin \theta)\rho d\rho d\theta</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.805em;"></span><span class="strut bottom" style="height:1.111em;vertical-align:-0.306em;"></span><span class="base textstyle uncramped"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0004999999999999727em;">∬</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">ρ</span><span class="mop">cos</span><span class="mord mathit" style="margin-right:0.02778em;">θ</span><span class="mpunct">,</span><span class="mord mathit">ρ</span><span class="mop">sin</span><span class="mord mathit" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mord mathit">ρ</span><span class="mord mathit">d</span><span class="mord mathit">ρ</span><span class="mord mathit">d</span><span class="mord mathit" style="margin-right:0.02778em;">θ</span></span></span></span></p></li> <li><p>对于e的<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>x</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{x}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span>次幂，考虑换元  <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi><mo>=</mo><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>x</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">t=-\frac{1}{x}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord mathit">t</span><span class="mrel">=</span><span class="mord">−</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span></p></li> <li><p>对于 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>f</mi><mo>(</mo><msub><mi>x</mi><mi>n</mi></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">x_{n+1} = f(x_n)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">n</span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span></span></span></span>，结合 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>y</mi><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">y = x</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="mrel">=</span><span class="mord mathit">x</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">y=f(x)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="mrel">=</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span>的图像做数学分析，可以判断 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_n</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>能否逼近极限，以及是否 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>{</mo><msub><mi>x</mi><mi>n</mi></msub><mo>}</mo></mrow><annotation encoding="application/x-tex">\{x_n\}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">{</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">}</span></span></span></span> 单调有界，如果单调，则极限等于 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">x = f(x)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit">x</span><span class="mrel">=</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span>的解，先在草稿纸上直接求出极限A， 然后在答题卡上证明 |x_n -A| \to 0(n\to +\infin) 。（注意不是单调有界不代表没有极限），为了证明该式子，一般需要利用 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>A</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">A= f(A)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit">A</span><span class="mrel">=</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">A</span><span class="mclose">)</span></span></span></span> 和 关于<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_n</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>的不等式两个条件，可以使用拉格朗日定理构造<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">∣</mi><msub><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><mi>A</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mi mathvariant="normal">∣</mi><mi>f</mi><mo>(</mo><msub><mi>x</mi><mi>n</mi></msub><mo>)</mo><mo>−</mo><mi>f</mi><mo>(</mo><mi>A</mi><mo>)</mo><mi mathvariant="normal">∣</mi><mo>=</mo><mi mathvariant="normal">∣</mi><msup><mi>f</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><mi>ξ</mi><mo>)</mo><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><msub><mi>x</mi><mi>n</mi></msub><mo>−</mo><mi>A</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|x_{n+1}-A|= |f(x_n) - f(A)| = |f'(\xi)||x_n - A|</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.751892em;"></span><span class="strut bottom" style="height:1.001892em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">∣</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">n</span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">−</span><span class="mord mathit">A</span><span class="mord mathrm">∣</span><span class="mrel">=</span><span class="mord mathrm">∣</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mbin">−</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">A</span><span class="mclose">)</span><span class="mord mathrm">∣</span><span class="mrel">=</span><span class="mord mathrm">∣</span><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span><span class="mord mathrm">∣</span><span class="mord mathrm">∣</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">−</span><span class="mord mathit">A</span><span class="mord mathrm">∣</span></span></span></span>的关系,然后进行放缩</p></li> <li><p>如果 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>{</mo><msub><mi>a</mi><mi>n</mi></msub><mo>}</mo></mrow><annotation encoding="application/x-tex">\{a_n\}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">{</span><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">}</span></span></span></span>单调递增，那么，有上界 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.36687em;"></span><span class="strut bottom" style="height:0.36687em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mrel">⇒</span></span></span></span> 有极限且等于上确界，无上界 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.36687em;"></span><span class="strut bottom" style="height:0.36687em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mrel">⇒</span></span></span></span>  a_n = +\infin</p></li> <li><p>分段函数x取值区间只在 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>[</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo separator="true">,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mo>)</mo></mrow><annotation encoding="application/x-tex">[\frac{1}{n+1},\frac{1}{n})</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.2484389999999999em;vertical-align:-0.403331em;"></span><span class="base textstyle uncramped"><span class="mopen">[</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">n</span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mpunct">,</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">n</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mclose">)</span></span></span></span>的，x趋于0的极限直接代入函数值计算，因为在 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>[</mo><mn>0</mn><mo separator="true">,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>]</mo></mrow><annotation encoding="application/x-tex">[0,\frac{1}{n+1}]</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.2484389999999999em;vertical-align:-0.403331em;"></span><span class="base textstyle uncramped"><span class="mopen">[</span><span class="mord mathrm">0</span><span class="mpunct">,</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">n</span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mclose">]</span></span></span></span>内没有定义，只能用夹逼准则做</p></li> <li><p>求解微分方程一定要检查是否存在隐含初值使得特解可求</p></li> <li><p>求函数导数用定义法 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>f</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>)</mo><mo>=</mo><msub><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><msub><mi>x</mi><mn>0</mn></msub></mrow></msub><mfrac><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>f</mi><mo>(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>)</mo></mrow><mrow><mi>x</mi><mo>−</mo><msub><mi>x</mi><mn>0</mn></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">f'(x_0) = \lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.01em;"></span><span class="strut bottom" style="height:1.46em;vertical-align:-0.44999999999999996em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mrel">=</span><span class="mop"><span class="mop">lim</span><span class="vlist"><span style="top:0.15000000000000002em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span><span class="mrel">→</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.07142857142857144em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathrm">0</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span><span class="mbin">−</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.07142857142857144em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathrm">0</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.485em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mbin">−</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.07142857142857144em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathrm">0</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span>,特别的，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>f</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></msub><mfrac><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>f</mi><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mrow><mi>x</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">f'(0) = \lim_{x\to 0} \frac{f(x)-f(0)}{x}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.01em;"></span><span class="strut bottom" style="height:1.355em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathrm">0</span><span class="mclose">)</span><span class="mrel">=</span><span class="mop"><span class="mop">lim</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span><span class="mrel">→</span><span class="mord mathrm">0</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.485em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mbin">−</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathrm">0</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span></p></li> <li><p>利用绝对值不等式可以证明，若 \lim|f(x)-A|存在\Rightarrow \lim |f(x)| = |A|</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>lim</mi><mi mathvariant="normal">∣</mi><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi mathvariant="normal">∣</mi><mo>=</mo><mn>0</mn><mo>⇔</mo><mi>lim</mi><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\lim |f(x)| = 0 \Leftrightarrow \lim f(x) = 0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mop">lim</span><span class="mord mathrm">∣</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mord mathrm">∣</span><span class="mrel">=</span><span class="mord mathrm">0</span><span class="mrel">⇔</span><span class="mop">lim</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathrm">0</span></span></span></span></p></li> <li><p>\lim_{x\to +\infin} P_n(x)e^{-x} =0,<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>P</mi><mi>n</mi></msub><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">P_n(x)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">P</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.13889em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span>是关于x的n次多项式</p></li> <li><p>(abc)’ = a’bc+ab’c+abc‘，前提是 a’ ,b’,c’存在，如果在特殊点导数不存在，则需要用定义法求(abc)’在特殊点的导数</p></li> <li><p>比较 \Delta y和dy大小要结合取值的正负号来判断</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>A</mi><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">A^T</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8413309999999999em;"></span><span class="strut bottom" style="height:0.8413309999999999em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">A</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">T</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>特征值与A相同，但特征向量没有联系</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>ξ</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>ξ</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\xi_1,\xi_2</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.04601em;">ξ</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.04601em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathit" style="margin-right:0.04601em;">ξ</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.04601em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>分别是不同特征值的特征向量，则 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>k</mi><mn>1</mn></msub><msub><mi>ξ</mi><mn>1</mn></msub><mo>+</mo><msub><mi>k</mi><mn>2</mn></msub><msub><mi>ξ</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>(</mo><mi mathvariant="normal">∣</mi><msub><mi>k</mi><mn>1</mn></msub><mi mathvariant="normal">∣</mi><mo>+</mo><mi mathvariant="normal">∣</mi><msub><mi>k</mi><mn>2</mn></msub><mi mathvariant="normal">∣</mi><mo>≠</mo><mn>0</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">k_1\xi_1+k_2\xi_2, (|k_1| + |k_2| \ne 0)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03148em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.04601em;">ξ</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.04601em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03148em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.04601em;">ξ</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.04601em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mpunct">,</span><span class="mopen">(</span><span class="mord mathrm">∣</span><span class="mord"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03148em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathrm">∣</span><span class="mbin">+</span><span class="mord mathrm">∣</span><span class="mord"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03148em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathrm">∣</span><span class="mrel">≠</span><span class="mord mathrm">0</span><span class="mclose">)</span></span></span></span>不是特征向量</p></li> <li><p>驻点是导数为0的点</p></li> <li><p>做选择题看到矩阵相加时要确定矩阵的阶是否相同，如果不同，则不能相加，也就错误</p></li> <li><p>如果求在某点的导数时，函数很复杂，不要求出导数后再求值，而是考虑用定义法直接求值</p></li> <li><p>奇函数在原点对称区间的定积分为0</p></li> <li><p>如果各阶偏导数都连续，故不同次序的混合偏导数相等</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>a</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{|x|}{x^2+a}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.01em;"></span><span class="strut bottom" style="height:1.413331em;vertical-align:-0.403331em;"></span><span class="base textstyle uncramped"><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.289em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord mathit">a</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.485em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">∣</span><span class="mord mathit">x</span><span class="mord mathrm">∣</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span>想到 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mo>+</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi>X</mi><mi mathvariant="normal">∣</mi></mrow></mfrac></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{|x| + \frac{a}{|X|}}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.575108em;vertical-align:-0.73em;"></span><span class="base textstyle uncramped"><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">∣</span><span class="mord mathit">x</span><span class="mord mathrm">∣</span><span class="mbin">+</span><span class="mord reset-scriptstyle scriptstyle cramped"><span class="sizing reset-size5 size5 reset-scriptstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.37142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord scriptscriptstyle cramped"><span class="mord mathrm">∣</span><span class="mord mathit" style="margin-right:0.07847em;">X</span><span class="mord mathrm">∣</span></span></span></span><span style="top:-0.22142857142857142em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord scriptscriptstyle cramped"><span class="mord mathit">a</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-scriptstyle textstyle uncramped nulldelimiter"></span></span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span>,套基本不等式</p></li> <li><p>给出f（x）函数，如果感觉条件过少，考虑代特殊值，如果得到<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(x_0) = 0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathrm">0</span></span></span></span>， 考虑拉格朗日定理</p></li> <li><p>爪形行列式化将其他列倍减到第一列，是第一列只有一个元素不为0</p></li> <li><p>两三角形行列式其他行都减去第一行，化为爪形</p></li> <li><p>极限证明出现含绝对值的不等式，可能用到 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">∣</mi><mi>f</mi><mo>(</mo><msub><mi>x</mi><mi>n</mi></msub><mo>)</mo><mi mathvariant="normal">∣</mi><mo>≤</mo><mn>0</mn><mo>⇒</mo><mi>lim</mi><mi>f</mi><mo>(</mo><msub><mi>x</mi><mi>n</mi></msub><mo>)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">|f(x_n)| \le 0 \Rightarrow \lim f(x_n) = 0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">∣</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mord mathrm">∣</span><span class="mrel">≤</span><span class="mord mathrm">0</span><span class="mrel">⇒</span><span class="mop">lim</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathrm">0</span></span></span></span></p></li> <li><p>第二数学归纳法</p></li> <li><p>（1）当n=1和n=2时，命题成立；①</p> <p>（2）假设当n＜k时，命题成立；②</p> <p>（3）证明n=k时，命题也成立。③</p></li> <li><p>f(x)是偶函数，则 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mi>a</mi></mrow><mi>a</mi></msubsup><mfrac><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mn>1</mn></mrow></mfrac><mi>d</mi><mi>x</mi><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi>a</mi></mrow><mi>a</mi></msubsup><mfrac><mrow><msup><mi>e</mi><mi>x</mi></msup><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></mrow><mrow><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mn>1</mn></mrow></mfrac><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msubsup><mo>∫</mo><mrow><mo>−</mo><mi>a</mi></mrow><mrow><mi>a</mi></mrow></msubsup><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\int_{-a}^a \frac{f(x)}{e^x+1}dx = \int_{-a}^a \frac{e^x f(x))}{e^x+1} dx = \frac{1}{2}\int_{-a}^{a} f(x)dx</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.01em;"></span><span class="strut bottom" style="height:1.424451em;vertical-align:-0.414451em;"></span><span class="base textstyle uncramped"><span class="mop"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="vlist"><span style="top:0.35612em;margin-left:-0.19445em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord">−</span><span class="mord mathit">a</span></span></span></span><span style="top:-0.41900000000000004em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">a</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord"><span class="mord mathit">e</span><span class="vlist"><span style="top:-0.289em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathit">x</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.485em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mord mathit">d</span><span class="mord mathit">x</span><span class="mrel">=</span><span class="mop"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="vlist"><span style="top:0.35612em;margin-left:-0.19445em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord">−</span><span class="mord mathit">a</span></span></span></span><span style="top:-0.41900000000000004em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">a</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord"><span class="mord mathit">e</span><span class="vlist"><span style="top:-0.289em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathit">x</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.485em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord"><span class="mord mathit">e</span><span class="vlist"><span style="top:-0.363em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord mathit">x</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mord mathit">d</span><span class="mord mathit">x</span><span class="mrel">=</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mop"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="vlist"><span style="top:0.35612em;margin-left:-0.19445em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord">−</span><span class="mord mathit">a</span></span></span></span><span style="top:-0.41900000000000004em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">a</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mord mathit">d</span><span class="mord mathit">x</span></span></span></span></p></li> <li><p>选择题涉及到反常积分等于某值，求参数，说明反常积分收敛，先使用p级数比较判别法排除干扰项</p></li> <li><p>观察反常积分的敛散性，对于 \int_0^\infin f(x)dx，应拆成 \int_0^1 f(x)dx + \int_1^\infin f(x)dx分别算</p></li> <li>
\int_0^1 \frac{1}{x^p}dx = \begin{cases}
收敛&amp;, 0 &lt; p &lt; 1\\
发散&amp;, p \ge 1
\end{cases}
\\
\int_1^\infin \frac{1}{x^p}dx = \begin{cases}
收敛&amp;, p &gt; 1\\
发散&amp;, p \le 1
\end{cases}

</li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>→</mo><mn>0</mn><mo separator="true">,</mo><msup><mi>x</mi><mn>2</mn></msup><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow><annotation encoding="application/x-tex">x \to 0,x^2 \to 0^+</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:1.008548em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit">x</span><span class="mrel">→</span><span class="mord mathrm">0</span><span class="mpunct">,</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">→</span><span class="mord"><span class="mord mathrm">0</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord">+</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>C</mi><mo>+</mo><mi>x</mi></mrow></mfrac><msup><mo>)</mo><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>=</mo><mo>(</mo><mo>−</mo><mn>1</mn><msup><mo>)</mo><mi>n</mi></msup><mfrac><mrow><mi>n</mi><mo>!</mo></mrow><mrow><mo>(</mo><mi>C</mi><mo>+</mo><mi>x</mi><msup><mo>)</mo><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">(\frac{1}{C+x})^{(n)}=(-1)^n\frac{n!}{(C+x)^{n+1}}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8879999999999999em;"></span><span class="strut bottom" style="height:1.408em;vertical-align:-0.52em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="mbin">+</span><span class="mord mathit">x</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mopen">(</span><span class="mord mathit">n</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mopen">(</span><span class="mord">−</span><span class="mord mathrm">1</span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.34500000000000003em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="mbin">+</span><span class="mord mathit">x</span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.289em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord scriptscriptstyle cramped"><span class="mord mathit">n</span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">n</span><span class="mclose">!</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>sin</mi><mi>k</mi><mi>x</mi><msup><mo>)</mo><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>=</mo><msup><mi>k</mi><mi>n</mi></msup><mi>sin</mi><mo>(</mo><mi>x</mi><mo>+</mo><mi>n</mi><mo>⋅</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><annotation encoding="application/x-tex">(\sin kx)^{(n)}=k^n\sin(x+n\cdot \frac{\pi}{2})</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8879999999999999em;"></span><span class="strut bottom" style="height:1.2329999999999999em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mop">sin</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mord mathit">x</span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mopen">(</span><span class="mord mathit">n</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mop">sin</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mbin">+</span><span class="mord mathit">n</span><span class="mbin">⋅</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.03588em;">π</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mclose">)</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mi>ln</mi><mn>4</mn><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\int_0^1 \ln (1+x)dx=\ln 4 -1</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.870108em;"></span><span class="strut bottom" style="height:1.2262279999999999em;vertical-align:-0.35612em;"></span><span class="base textstyle uncramped"><span class="mop"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="vlist"><span style="top:0.35612em;margin-left:-0.19445em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span style="top:-0.41900000000000004em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord mathrm">1</span><span class="mbin">+</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mord mathit">d</span><span class="mord mathit">x</span><span class="mrel">=</span><span class="mop">ln</span><span class="mord mathrm">4</span><span class="mbin">−</span><span class="mord mathrm">1</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><msup><mi>a</mi><mi>x</mi></msup><msup><mo>)</mo><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>=</mo><msup><mi>a</mi><mi>x</mi></msup><msup><mi>ln</mi><mi>n</mi></msup><mi>a</mi></mrow><annotation encoding="application/x-tex">(a^x)^{(n)}=a^x\ln^n a</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8879999999999999em;"></span><span class="strut bottom" style="height:1.138em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">x</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mopen">(</span><span class="mord mathit">n</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">x</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mop"><span class="mop">ln</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit">a</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>cos</mi><mi>k</mi><mi>x</mi><msup><mo>)</mo><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>=</mo><msup><mi>k</mi><mi>n</mi></msup><mi>cos</mi><mo>(</mo><mi>x</mi><mo>+</mo><mi>n</mi><mo>⋅</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><annotation encoding="application/x-tex">(\cos kx)^{(n)}=k^n\cos(x+n\cdot \frac{\pi}{2})</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8879999999999999em;"></span><span class="strut bottom" style="height:1.2329999999999999em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mop">cos</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mord mathit">x</span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mopen">(</span><span class="mord mathit">n</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mbin">+</span><span class="mord mathit">n</span><span class="mbin">⋅</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.03588em;">π</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mclose">)</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>2</mn><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>⋅</mo><msup><mi>f</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>[</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>⋅</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><msup><mo>]</mo><mrow><mi mathvariant="normal">′</mi></mrow></msup></mrow><annotation encoding="application/x-tex">2f(x)\cdot f'(x) = [f(x)\cdot f(x)]'</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.751892em;"></span><span class="strut bottom" style="height:1.001892em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">2</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mbin">⋅</span><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mrel">=</span><span class="mopen">[</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mbin">⋅</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mclose"><span class="mclose">]</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><msup><mi>f</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{f'(x)}{f(x)}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.01688em;"></span><span class="strut bottom" style="height:1.53688em;vertical-align:-0.52em;"></span><span class="base textstyle uncramped"><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.34500000000000003em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.485em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:-0.363em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord scriptscriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span>考虑 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>ln</mi><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">F(x) = \ln f(x)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mrel">=</span><span class="mop">ln</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span></p></li> <li><p>$[f'(x)]^2 +f(x)\cdot f''(x) = [f(x)\cdot f'(x)]' $</p></li> <li><p>f'(x)+\phi'(x)f(x),考虑 f(x)\cdot e^{\phi(x)}</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>u</mi><mi>v</mi><msup><mo>)</mo><mrow><mi mathvariant="normal">′</mi><mi mathvariant="normal">′</mi></mrow></msup><mo>=</mo><msup><mi>u</mi><mrow><mi mathvariant="normal">′</mi><mi mathvariant="normal">′</mi></mrow></msup><mo>+</mo><mn>2</mn><msup><mi>u</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><msup><mi>v</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>+</mo><msup><mi>v</mi><mrow><mi mathvariant="normal">′</mi><mi mathvariant="normal">′</mi></mrow></msup></mrow><annotation encoding="application/x-tex">(uv)'' = u''+2u'v'+v''</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.751892em;"></span><span class="strut bottom" style="height:1.001892em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathit">u</span><span class="mord mathit" style="margin-right:0.03588em;">v</span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord"><span class="mord mathit">u</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord mathrm">2</span><span class="mord"><span class="mord mathit">u</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">v</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">v</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></p></li> <li><p>涉及到 f'(x)与f(x)的关系，都可以考虑拉格朗日中值定理</p></li> <li><p>非零对角矩阵的相乘不会改变其非零性</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>A</mi><mn>3</mn></msup><mo>−</mo><mi>E</mi><mo>=</mo><mo>(</mo><mi>A</mi><mo>−</mo><mi>E</mi><mo>)</mo><mo>(</mo><msup><mi>A</mi><mn>2</mn></msup><mo>+</mo><mi>A</mi><mo>+</mo><mi>E</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">A^3-E = (A-E)(A^2+A+E)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">A</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">−</span><span class="mord mathit" style="margin-right:0.05764em;">E</span><span class="mrel">=</span><span class="mopen">(</span><span class="mord mathit">A</span><span class="mbin">−</span><span class="mord mathit" style="margin-right:0.05764em;">E</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">A</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord mathit">A</span><span class="mbin">+</span><span class="mord mathit" style="margin-right:0.05764em;">E</span><span class="mclose">)</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>A</mi><mn>3</mn></msup><mo>+</mo><mi>E</mi><mo>=</mo><mo>(</mo><mi>A</mi><mo>+</mo><mi>E</mi><mo>)</mo><mo>(</mo><msup><mi>A</mi><mn>2</mn></msup><mo>−</mo><mi>A</mi><mo>+</mo><mi>E</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">A^3+E=(A+E)(A^2-A+E)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">A</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">3</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord mathit" style="margin-right:0.05764em;">E</span><span class="mrel">=</span><span class="mopen">(</span><span class="mord mathit">A</span><span class="mbin">+</span><span class="mord mathit" style="margin-right:0.05764em;">E</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">A</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">−</span><span class="mord mathit">A</span><span class="mbin">+</span><span class="mord mathit" style="margin-right:0.05764em;">E</span><span class="mclose">)</span></span></span></span></p></li> <li><p>矩阵等价具有传递性</p></li> <li><p>对 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msubsup><mi>f</mi><mo>(</mo><mi>sin</mi><mi>x</mi><mo>)</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>d</mi><mi>x</mi><mo>≥</mo><msubsup><mo>∫</mo><mn>0</mn><mrow><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msubsup><mi>f</mi><mo>(</mo><mi>cos</mi><mi>x</mi><mo>)</mo><mo>)</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\int_0^{\frac{\pi}{2}}f(\sin x)g(x)dx \ge \int_0^{\frac{\pi}{2}} f(\cos x))g(x)dx</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.07738em;"></span><span class="strut bottom" style="height:1.3835em;vertical-align:-0.30612em;"></span><span class="base textstyle uncramped"><span class="mop"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="vlist"><span style="top:0.266308em;margin-left:-0.19445em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span style="top:-0.5862999999999999em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord reset-scriptstyle scriptstyle uncramped"><span class="sizing reset-size5 size5 reset-scriptstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord scriptscriptstyle cramped"><span class="mord mathrm">2</span></span></span></span><span style="top:-0.22142857142857142em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord scriptscriptstyle uncramped"><span class="mord mathit" style="margin-right:0.03588em;">π</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-scriptstyle textstyle uncramped nulldelimiter"></span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mop">sin</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mord mathit">d</span><span class="mord mathit">x</span><span class="mrel">≥</span><span class="mop"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="vlist"><span style="top:0.266308em;margin-left:-0.19445em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span style="top:-0.5862999999999999em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord reset-scriptstyle scriptstyle uncramped"><span class="sizing reset-size5 size5 reset-scriptstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord scriptscriptstyle cramped"><span class="mord mathrm">2</span></span></span></span><span style="top:-0.22142857142857142em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord scriptscriptstyle uncramped"><span class="mord mathit" style="margin-right:0.03588em;">π</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-scriptstyle textstyle uncramped nulldelimiter"></span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mop">cos</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mclose">)</span><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mord mathit">d</span><span class="mord mathit">x</span></span></span></span>的证明问题，令左式-右式，积分区域拆成 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>0</mn><mo separator="true">,</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo>)</mo></mrow><annotation encoding="application/x-tex">(0,\frac{\pi}{4})</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1.095em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathrm">0</span><span class="mpunct">,</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">4</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.03588em;">π</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mclose">)</span></span></span></span>和 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo separator="true">,</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><annotation encoding="application/x-tex">(\frac{\pi}{4}, \frac{\pi}{2})</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1.095em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">4</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.03588em;">π</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mpunct">,</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.03588em;">π</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mclose">)</span></span></span></span>两个部分，令用 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi><mo>=</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">t = \frac{\pi}{2}-x</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.695392em;"></span><span class="strut bottom" style="height:1.040392em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord mathit">t</span><span class="mrel">=</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.03588em;">π</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mbin">−</span><span class="mord mathit">x</span></span></span></span>替换其中一项，从而得到一个可以判断正负的定积分，原不等式得证</p></li> <li><p>可导必连续，连续未必可导；判断是否连续看左极限是否等于右极限，是否可导需要用导数的定义法看 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>)</mo><mo>−</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{f(x+\Delta x)- f(x)}{\Delta x}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.01em;"></span><span class="strut bottom" style="height:1.355em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">Δ</span><span class="mord mathit">x</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.485em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mbin">+</span><span class="mord mathrm">Δ</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mbin">−</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span>的极限是否存在</p></li> <li><p>求 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mi>x</mi><mi>a</mi></msup><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">g(x) = x^af(x)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mrel">=</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">a</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span> 的n阶导在x=0的结果，分别对 f(x)和 g(x) 使用泰勒公式，然后令逐项相等，得出 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>g</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>(</mo><mn>0</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">g^{(n)}(0)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8879999999999999em;"></span><span class="strut bottom" style="height:1.138em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">g</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mopen">(</span><span class="mord mathit">n</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathrm">0</span><span class="mclose">)</span></span></span></span>的值</p></li> <li><p>泰勒展开式中，奇函数展开不会有偶数次方，偶函数展开不会有奇数次方的性质</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>f</mi><mrow><mi mathvariant="normal">′</mi><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">f''(0)=A</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.751892em;"></span><span class="strut bottom" style="height:1.001892em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathrm">0</span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathit">A</span></span></span></span>只能得出在x=0的邻域里一阶可导，研究 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>f</mi><mrow><mi mathvariant="normal">′</mi><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">f''(x)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.751892em;"></span><span class="strut bottom" style="height:1.001892em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span>在x=0的情形仍需要用定义法，并且要求x在x=0的邻域里</p></li> <li><p>如果 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span>在x=0处连续， 且 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></msub><mfrac><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>x</mi></mrow></mfrac><mo>=</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\lim_{x\to 0} \frac{f(x)}{x} =A</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.01em;"></span><span class="strut bottom" style="height:1.355em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mop"><span class="mop">lim</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span><span class="mrel">→</span><span class="mord mathrm">0</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">x</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.485em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mrel">=</span><span class="mord mathit">A</span></span></span></span>，则 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn><mo separator="true">,</mo><msup><mi>f</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">f(0)=0, f'(0) = A</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.751892em;"></span><span class="strut bottom" style="height:1.001892em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathrm">0</span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathrm">0</span><span class="mpunct">,</span><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathrm">0</span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathit">A</span></span></span></span></p></li> <li><p>零点定理要求函数连续，导数零点定理只要求函数可导（<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>f</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><mi>a</mi><mo>)</mo><msup><mi>f</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><mi>b</mi><mo>)</mo><mo>&lt;</mo><mn>0</mn><mo>⇒</mo><msup><mi>f</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><mi>c</mi><mo>)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f'(a)f'(b)&lt;0 \Rightarrow f'(c)=0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.751892em;"></span><span class="strut bottom" style="height:1.001892em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit">a</span><span class="mclose">)</span><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit">b</span><span class="mclose">)</span><span class="mrel">&lt;</span><span class="mord mathrm">0</span><span class="mrel">⇒</span><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit">c</span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathrm">0</span></span></span></span>)</p></li> <li><p>拉格朗日中值定理要求闭区间连续，开区间可导</p></li> <li><p>证明至少有一根：1、零点定理 2、罗尔定理证明<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>F</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>=</mo><mi>F</mi><mo>(</mo><mi>b</mi><mo>)</mo><mo>⇒</mo><msup><mi>F</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">F(a)=F(b)\Rightarrow F'(x) = f(x) = 0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.751892em;"></span><span class="strut bottom" style="height:1.001892em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathit">a</span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathit">b</span><span class="mclose">)</span><span class="mrel">⇒</span><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathrm">0</span></span></span></span></p></li> <li><p>曲率k = \frac{|y''|}{(1+y'^2)^{\frac{3}{2}}} = \frac{\phi ''(t)\psi'(t) - \phi'(t)\psi''(t)}{[\phi'^2(t)+\psi'^2(t)]^{\frac{3}{2}}}</p></li> <li><p>广义p积分</p></li> <li><p>拉格朗日中值定理可看成是带拉格朗日余项的泰勒公式的特例，拉格朗日中值定理解决不了的，考虑泰勒公式</p></li> <li><p>证明恒等式，构造F（x）=0，证 F‘（x） = 0</p></li> <li><p>曲率圆的圆心坐标为 $(x_0= a - f'(a)\frac{1+f'^2(a)}{f''(a)} ,y_0 = b + \frac{1+f'^2(a)}{f''(a)}) $</p></li> <li><p>题目只给了一个不等式和某个初值，求某个值，可以利用递推关系求极限值</p></li> <li><p>遇到包含 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{1+x^2}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.2484389999999999em;vertical-align:-0.403331em;"></span><span class="base textstyle uncramped"><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">1</span><span class="mbin">+</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:-0.289em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span>的积分，可以考虑换元 令 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>=</mo><mi>tan</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">x = \tan t</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.61508em;"></span><span class="strut bottom" style="height:0.61508em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">x</span><span class="mrel">=</span><span class="mop">tan</span><span class="mord mathit">t</span></span></span></span></p></li> <li><p>若r(A) = 1, 则 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>λ</mi><mn>1</mn></msub><mo>=</mo><mo>⋯</mo><mo>=</mo><msub><mi>λ</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>0</mn><mo separator="true">,</mo><msub><mi>λ</mi><mi>n</mi></msub><mo>=</mo><mi>t</mi><mi>r</mi><mo>(</mo><mi>A</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\lambda_1 = \cdots = \lambda_{n-1} = 0, \lambda_n = tr(A)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">λ</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="minner">⋯</span><span class="mrel">=</span><span class="mord"><span class="mord mathit">λ</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">n</span><span class="mbin">−</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord mathrm">0</span><span class="mpunct">,</span><span class="mord"><span class="mord mathit">λ</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord mathit">t</span><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="mopen">(</span><span class="mord mathit">A</span><span class="mclose">)</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mi>f</mi><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mrow><mi>ξ</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{f(\xi)}{\xi}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.01em;"></span><span class="strut bottom" style="height:1.491108em;vertical-align:-0.481108em;"></span><span class="base textstyle uncramped"><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit" style="margin-right:0.04601em;">ξ</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.485em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span>考虑柯西中值定理，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mi>f</mi><mo>(</mo><mi>b</mi><mo>)</mo><mo>−</mo><mi>f</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{f(b)-f(a)}{b^2-a^2}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.01em;"></span><span class="strut bottom" style="height:1.413331em;vertical-align:-0.403331em;"></span><span class="base textstyle uncramped"><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord"><span class="mord mathit">b</span><span class="vlist"><span style="top:-0.289em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">−</span><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:-0.289em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.485em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">b</span><span class="mclose">)</span><span class="mbin">−</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">a</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span></p></li> <li><p>由介值定理必然存在 f(\xi) = \frac{f(a)+f(b)}{2}， a &lt; \xi &lt; b</p></li> <li><p>费马定理的应用，在(a,b)区间内找到一点 \xi，使f(\xi)&gt;f(a),f(\xi)&gt;f(b)，说明极大值在 (a,b)内取得，则存在 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>f</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><mi>η</mi><mo>)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f'(\eta)=0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.751892em;"></span><span class="strut bottom" style="height:1.001892em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.03588em;">η</span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathrm">0</span></span></span></span></p></li> <li><p>如果 g''(x)\ne0，g(a)=g(b)，则在(a,b)内处处 g(x)\ne g(a)或g(x) \ne g(b)</p></li> <li><p>泰勒公式的另一种形式（用于寻找各阶导数的关系）<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>+</mo><mi>h</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><msup><mi>f</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mi>h</mi><mo>+</mo><mfrac><mrow><msup><mi>f</mi><mrow><mi mathvariant="normal">′</mi><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><msup><mi>h</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mo>+</mo><mo>⋯</mo><mo>+</mo><mfrac><mrow><msup><mi>f</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>(</mo><mi>ξ</mi><mo>)</mo><msup><mi>h</mi><mi>n</mi></msup></mrow><mrow><mi>n</mi><mo>!</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">f(x+h) = f(x)+f'(x)h+\frac{f''(x)h^2}{2!}+\cdots+\frac{f^{(n)}(\xi)h^n}{n!}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.1141em;"></span><span class="strut bottom" style="height:1.4591em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mbin">+</span><span class="mord mathit">h</span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mbin">+</span><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mord mathit">h</span><span class="mbin">+</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span><span class="mclose">!</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.485em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:-0.363em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord scriptscriptstyle uncramped"><span class="mord mathrm">′</span><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mord"><span class="mord mathit">h</span><span class="vlist"><span style="top:-0.363em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mbin">+</span><span class="minner">⋯</span><span class="mbin">+</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">n</span><span class="mclose">!</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.485em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:-0.363em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord scriptscriptstyle uncramped"><span class="mopen">(</span><span class="mord mathit">n</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span><span class="mord"><span class="mord mathit">h</span><span class="vlist"><span style="top:-0.363em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span></p></li> <li><p>如果一阶导数不便确定正负，可尝试计算二阶导，根据单调性和驻点确定极值，从而得到不等式</p></li> <li><p><img src="/assets/img/image-20201019102358060.7118368e.png" alt="image-20201019102358060"></p></li> <li><p>对于对偶式的不等式证明， 令其中一个参数为x，构造函数研究单调性</p></li> <li><p>如果变限积分包含了积分限变量，需要换元消去该变量</p></li> <li><p>证明积分不等式，使用下限变量化，构造函数证明</p></li> <li><p><img src="/assets/img/image-20200926151058929.422f4bd6.png" alt="image-20200926151058929"></p></li> <li><p>上题使用积分中值定理，二重积分的其中一个积分变量没有出现在另一个积分的积分限上，则可以考虑积分中值定理</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>π</mi></msubsup><mfrac><mrow><mi>x</mi><mi>sin</mi><mi>x</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>sin</mi><mn>2</mn></msup><mi>x</mi></mrow></mfrac><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\int_0^\pi \frac{x\sin x}{1+\sin^2 x}dx</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.861502em;"></span><span class="strut bottom" style="height:1.2648329999999999em;vertical-align:-0.403331em;"></span><span class="base textstyle uncramped"><span class="mop"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="vlist"><span style="top:0.35612em;margin-left:-0.19445em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span style="top:-0.41900000000000004em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.03588em;">π</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">1</span><span class="mbin">+</span><span class="mop"><span class="mop">sin</span><span class="vlist"><span style="top:-0.289em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit">x</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">x</span><span class="mop">sin</span><span class="mord mathit">x</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mord mathit">d</span><span class="mord mathit">x</span></span></span></span> 令 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>=</mo><mi>π</mi><mo>−</mo><mi>u</mi></mrow><annotation encoding="application/x-tex">x = \pi - u</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.58333em;"></span><span class="strut bottom" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="base textstyle uncramped"><span class="mord mathit">x</span><span class="mrel">=</span><span class="mord mathit" style="margin-right:0.03588em;">π</span><span class="mbin">−</span><span class="mord mathit">u</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>a</mi></msubsup><mi>f</mi><mo>(</mo><mi>a</mi><mo>−</mo><mi>t</mi><mo>)</mo><mi>d</mi><mi>t</mi><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi>a</mi></msubsup><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\int_0^a f(a-t)dt = \int_0^a f(t)dt</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.805em;"></span><span class="strut bottom" style="height:1.16112em;vertical-align:-0.35612em;"></span><span class="base textstyle uncramped"><span class="mop"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="vlist"><span style="top:0.35612em;margin-left:-0.19445em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span style="top:-0.41900000000000004em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">a</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">a</span><span class="mbin">−</span><span class="mord mathit">t</span><span class="mclose">)</span><span class="mord mathit">d</span><span class="mord mathit">t</span><span class="mrel">=</span><span class="mop"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="vlist"><span style="top:0.35612em;margin-left:-0.19445em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span style="top:-0.41900000000000004em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">a</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">t</span><span class="mclose">)</span><span class="mord mathit">d</span><span class="mord mathit">t</span></span></span></span></p></li> <li><p>遇到复杂的定积分，优先判断是不是奇函数，能否化简</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>∫</mo><mi>sec</mi><mi>t</mi><mi>d</mi><mi>t</mi><mo>=</mo><mo>∫</mo><mfrac><mrow><mi>sec</mi><mi>t</mi><mo>(</mo><mi>sec</mi><mi>t</mi><mo>+</mo><mi>tan</mi><mi>t</mi><mo>)</mo></mrow><mrow><mi>sec</mi><mi>t</mi><mo>+</mo><mi>tan</mi><mi>t</mi></mrow></mfrac><mi>d</mi><mi>t</mi><mo>=</mo><mo>∫</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>sec</mi><mi>t</mi><mo>+</mo><mi>tan</mi><mi>t</mi></mrow></mfrac><mi>d</mi><mo>(</mo><mi>sec</mi><mi>t</mi><mo>+</mo><mi>tan</mi><mi>t</mi><mo>)</mo><mo>=</mo><mi>ln</mi><mi mathvariant="normal">∣</mi><mi>sec</mi><mi>t</mi><mo>+</mo><mi>tan</mi><mi>t</mi><mi mathvariant="normal">∣</mi><mo>+</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\int \sec t dt = \int \frac{\sec t(\sec t +\tan t)}{\sec t + \tan t}dt = \int \frac{1}{\sec t + \tan t}d(\sec t + \tan t) = \ln | \sec t + \tan t| + C</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.01em;"></span><span class="strut bottom" style="height:1.413331em;vertical-align:-0.403331em;"></span><span class="base textstyle uncramped"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="mop">sec</span><span class="mord mathit">t</span><span class="mord mathit">d</span><span class="mord mathit">t</span><span class="mrel">=</span><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mop">sec</span><span class="mord mathit">t</span><span class="mbin">+</span><span class="mop">tan</span><span class="mord mathit">t</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.485em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mop">sec</span><span class="mord mathit">t</span><span class="mopen">(</span><span class="mop">sec</span><span class="mord mathit">t</span><span class="mbin">+</span><span class="mop">tan</span><span class="mord mathit">t</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mord mathit">d</span><span class="mord mathit">t</span><span class="mrel">=</span><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mop">sec</span><span class="mord mathit">t</span><span class="mbin">+</span><span class="mop">tan</span><span class="mord mathit">t</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mord mathit">d</span><span class="mopen">(</span><span class="mop">sec</span><span class="mord mathit">t</span><span class="mbin">+</span><span class="mop">tan</span><span class="mord mathit">t</span><span class="mclose">)</span><span class="mrel">=</span><span class="mop">ln</span><span class="mord mathrm">∣</span><span class="mop">sec</span><span class="mord mathit">t</span><span class="mbin">+</span><span class="mop">tan</span><span class="mord mathit">t</span><span class="mord mathrm">∣</span><span class="mbin">+</span><span class="mord mathit" style="margin-right:0.07153em;">C</span></span></span></span></p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>∫</mo><msup><mi>sec</mi><mi>n</mi></msup><mi>x</mi><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>(</mo><mi>tan</mi><mi>x</mi><msup><mi>sec</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>x</mi><mo>+</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>∫</mo><msup><mi>sec</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>d</mi><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\int \sec^n x dx = \frac{1}{n-1}(\tan x \sec^{n-2}x + (n-1)\int \sec^{n-2}dx)</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.845108em;"></span><span class="strut bottom" style="height:1.2484389999999999em;vertical-align:-0.403331em;"></span><span class="base textstyle uncramped"><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="mop"><span class="mop">sec</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit">x</span><span class="mord mathit">d</span><span class="mord mathit">x</span><span class="mrel">=</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">n</span><span class="mbin">−</span><span class="mord mathrm">1</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mopen">(</span><span class="mop">tan</span><span class="mord mathit">x</span><span class="mop"><span class="mop">sec</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">n</span><span class="mbin">−</span><span class="mord mathrm">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit">x</span><span class="mbin">+</span><span class="mopen">(</span><span class="mord mathit">n</span><span class="mbin">−</span><span class="mord mathrm">1</span><span class="mclose">)</span><span class="op-symbol small-op mop" style="margin-right:0.19445em;top:-0.0005599999999999772em;">∫</span><span class="mop"><span class="mop">sec</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">n</span><span class="mbin">−</span><span class="mord mathrm">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit">d</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span></p></li> <li><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><msub><mi>ϕ</mi><mn>1</mn></msub><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><msub><mi>ϕ</mi><mn>2</mn></msub><mo>(</mo><mi>x</mi><mo>)</mo></mrow></msubsup><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo><mi>d</mi><mi>t</mi><mo separator="true">,</mo><msup><mi>F</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>[</mo><msub><mi>ϕ</mi><mn>2</mn></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>]</mo><msubsup><mi>ϕ</mi><mn>2</mn><mrow><mi mathvariant="normal">′</mi></mrow></msubsup><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>f</mi><mo>[</mo><msub><mi>ϕ</mi><mn>1</mn></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>]</mo><msubsup><mi>ϕ</mi><mn>1</mn><mrow><mi mathvariant="normal">′</mi></mrow></msubsup><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">F(x)=\int_{\phi_1(x)}^{\phi_2(x)} f(t)dt,
F'(x) = f[\phi_2(x)]\phi_2'(x)- f[\phi_1(x) ]\phi_1'(x)
</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.499em;"></span><span class="strut bottom" style="height:2.58625em;vertical-align:-1.08725em;"></span><span class="base displaystyle textstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mrel">=</span><span class="mop"><span class="op-symbol large-op mop" style="margin-right:0.44445em;top:-0.0011249999999999316em;">∫</span><span class="vlist"><span style="top:0.91225em;margin-left:-0.44445em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord"><span class="mord mathit">ϕ</span><span class="vlist"><span style="top:0.15em;margin-right:0.07142857142857144em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathrm">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span><span style="top:-0.9740000000000002em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord"><span class="mord mathit">ϕ</span><span class="vlist"><span style="top:0.15em;margin-right:0.07142857142857144em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">t</span><span class="mclose">)</span><span class="mord mathit">d</span><span class="mord mathit">t</span><span class="mpunct">,</span><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="vlist"><span style="top:-0.413em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">[</span><span class="mord"><span class="mord mathit">ϕ</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mclose">]</span><span class="mord"><span class="mord mathit">ϕ</span><span class="vlist"><span style="top:0.247em;margin-left:0em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">2</span></span></span><span style="top:-0.4129999999999999em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mbin">−</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">[</span><span class="mord"><span class="mord mathit">ϕ</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mclose">]</span><span class="mord"><span class="mord mathit">ϕ</span><span class="vlist"><span style="top:0.247em;margin-left:0em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">1</span></span></span><span style="top:-0.4129999999999999em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">′</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span></span></span></span></span></p></li> <li><p>抽象矩阵求行列式可以考虑使用特征值来求</p></li> <li><p>副对角线行列式 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">∣</mi><mi>A</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mo>(</mo><mo>−</mo><mn>1</mn><msup><mo>)</mo><mrow><mfrac><mrow><mi>n</mi><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac><mo>∏</mo><msub><mi>a</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>i</mi></mrow></msub></mrow></msup></mrow><annotation encoding="application/x-tex">|A| = (-1)^{\frac{n(n-1)}{2} \prod a_{i, n+1-i}}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.0979999999999999em;"></span><span class="strut bottom" style="height:1.3479999999999999em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">∣</span><span class="mord mathit">A</span><span class="mord mathrm">∣</span><span class="mrel">=</span><span class="mopen">(</span><span class="mord">−</span><span class="mord mathrm">1</span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord reset-scriptstyle scriptstyle uncramped"><span class="sizing reset-size5 size5 reset-scriptstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord scriptscriptstyle cramped"><span class="mord mathrm">2</span></span></span></span><span style="top:-0.22142857142857142em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle textstyle uncramped frac-line"></span></span><span style="top:-0.5142857142857143em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord scriptscriptstyle uncramped"><span class="mord mathit">n</span><span class="mopen">(</span><span class="mord mathit">n</span><span class="mbin">−</span><span class="mord mathrm">1</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-scriptstyle textstyle uncramped nulldelimiter"></span></span><span class="op-symbol small-op mop" style="top:0.074995em;">∏</span><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:0.15000000000000002em;margin-right:0.07142857142857144em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord scriptscriptstyle cramped"><span class="mord mathit">i</span><span class="mpunct">,</span><span class="mord mathit">n</span><span class="mbin">+</span><span class="mord mathrm">1</span><span class="mbin">−</span><span class="mord mathit">i</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></p></li> <li><p>拉普拉斯展开式副对角线行列式 = <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mo>−</mo><mn>1</mn><msup><mo>)</mo><mrow><mi>m</mi><mi>n</mi></mrow></msup><mi mathvariant="normal">∣</mi><mi>A</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>B</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">(-1)^{mn}|A||B|</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord">−</span><span class="mord mathrm">1</span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">m</span><span class="mord mathit">n</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathrm">∣</span><span class="mord mathit">A</span><span class="mord mathrm">∣</span><span class="mord mathrm">∣</span><span class="mord mathit" style="margin-right:0.05017em;">B</span><span class="mord mathrm">∣</span></span></span></span></p></li> <li><p>范德蒙行列式可用第一数学归纳法证明</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>A</mi><msup><mi mathvariant="normal">∣</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>A</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mi mathvariant="normal">∣</mi><mi>A</mi><msup><mi mathvariant="normal">∣</mi><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><msup><mo>)</mo><mn>2</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">| |A|^{n-2}A | = |A|^{(n-1)^2}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.9393199999999999em;"></span><span class="strut bottom" style="height:1.18932em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">∣</span><span class="mord mathrm">∣</span><span class="mord mathit">A</span><span class="mord"><span class="mord mathrm">∣</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">n</span><span class="mbin">−</span><span class="mord mathrm">2</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit">A</span><span class="mord mathrm">∣</span><span class="mrel">=</span><span class="mord mathrm">∣</span><span class="mord mathit">A</span><span class="mord"><span class="mord mathrm">∣</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mopen">(</span><span class="mord mathit">n</span><span class="mbin">−</span><span class="mord mathrm">1</span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.363em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></p></li> <li><p>若 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi><mo>∼</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \sim B</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">A</span><span class="mrel">∼</span><span class="mord mathit" style="margin-right:0.05017em;">B</span></span></span></span>，则 |A| = |B|</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>A</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">A_{ij}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">A</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">i</span><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>是代数余子式， <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>M</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">M_{ij}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.10903em;">M</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.10903em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">i</span><span class="mord mathit" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>是余子式</p></li> <li><p>代数余子式乘积做差可以看做是伴随矩阵的子式</p></li> <li><p>主三角矩阵的主对角线的元素是特征值</p></li> <li><p>求解含参3元线性方程的唯一解时，建议使用克拉默法则求解</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>k</mi><mi>A</mi><msup><mo>)</mo><mo>∗</mo></msup><mo>=</mo><msup><mi>k</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><msup><mi>A</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">(kA)^* = k^{n-1}A^*</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mord mathit">A</span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord">∗</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">n</span><span class="mbin">−</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord"><span class="mord mathit">A</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord">∗</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></p></li> <li><p>A不可逆等价于 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi><mi>ξ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">A\xi = 0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit">A</span><span class="mord mathit" style="margin-right:0.04601em;">ξ</span><span class="mrel">=</span><span class="mord mathrm">0</span></span></span></span></p></li> <li><p>实对称矩阵的不同特征值对应的特征向量两两正交</p></li> <li><p>当涉及到很多个三角函数相乘时，考虑凑微分成d(cos t)或d(sint t)，然后变成一个多项式积分</p></li> <li><p>A和A*具有相同的特征向量的前提是特征值不为0</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.0037em;">α</span></span></span></span>为 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>A</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">A^2</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">A</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>的特征向量， <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.0037em;">α</span></span></span></span>不一定为A的特征向量</p></li> <li><p>如果 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>λ</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>λ</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\lambda_1, \lambda_2</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">λ</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathit">λ</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>分别是A的不同的特征值，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_1,x_2</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>分别是对应的特征向量，则 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_1+x_2</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.58333em;"></span><span class="strut bottom" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mbin">+</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>一定不是A的特征向量</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>λ</mi><mi>E</mi><mo>−</mo><mi>A</mi><mo>)</mo><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">(\lambda E - A)x = 0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathit">λ</span><span class="mord mathit" style="margin-right:0.05764em;">E</span><span class="mbin">−</span><span class="mord mathit">A</span><span class="mclose">)</span><span class="mord mathit">x</span><span class="mrel">=</span><span class="mord mathrm">0</span></span></span></span>的解向量不一定是特征向量，因为特征向量不能是零向量</p></li> <li><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>a</mi><mi>B</mi><msup><mo>)</mo><mo>∗</mo></msup><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><msup><mi>B</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">(aB)^*=a^2B^*</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mopen">(</span><span class="mord mathit">a</span><span class="mord mathit" style="margin-right:0.05017em;">B</span><span class="mclose"><span class="mclose">)</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord">∗</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">=</span><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathrm">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.05017em;">B</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord">∗</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></p></li> <li><p>如果 r(A)=1, 则 A = <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>α</mi><mi>T</mi></msup><mi>β</mi></mrow><annotation encoding="application/x-tex">\alpha^T\beta</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8413309999999999em;"></span><span class="strut bottom" style="height:1.035771em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.0037em;">α</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">T</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.05278em;">β</span></span></span></span>, 设 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>b</mi><mn>1</mn></msub><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">b_1 \ne 0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.716em;"></span><span class="strut bottom" style="height:0.9309999999999999em;vertical-align:-0.215em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">b</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mrel">≠</span><span class="mord mathrm">0</span></span></span></span> , 则A的特征值是 0（n-1重), tr(A), 对应的特征向量分别是($[-b_2, b_1, 0, \cdots, 0]^T,[-b_3, 0, b_1,\cdots, 0]^T, \cdots ,[-b_n, 0, 0, \cdots, b_1], \alpha $)</p></li> <li><p>相似必合同，合同未必相似</p></li> <li><p>正定矩阵一定是实对称矩阵</p></li> <li><p>A与B合同，前提是A和B都是实对称矩阵</p></li> <li><p>正惯性指数要化成标准形才能知道</p></li> <li><p>规范形只有1，-1， 0</p></li> <li><p>顺序主子式是用于判断正定矩阵的</p></li> <li><p>利用特征值法判断二次型矩阵的正定性，可结合矩阵的迹和行列式，确定特征值的符号，从而做出判断</p></li> <li><p>正定矩阵的秩=正惯性指数+负惯性指数</p></li> <li><p>已知二次型的特征值和特征向量，求f(<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>)，可考虑用特征向量将<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span>线性表出，利用<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi><mi>ξ</mi><mo>=</mo><mi>λ</mi><mi>ξ</mi></mrow><annotation encoding="application/x-tex">A\xi = \lambda \xi</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit">A</span><span class="mord mathit" style="margin-right:0.04601em;">ξ</span><span class="mrel">=</span><span class="mord mathit">λ</span><span class="mord mathit" style="margin-right:0.04601em;">ξ</span></span></span></span>求值</p></li> <li><p>正定矩阵A必与单位矩阵合同，即存在可逆矩阵C，使<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi><mo>=</mo><msup><mi>C</mi><mi>T</mi></msup><mi>E</mi><mi>C</mi><mo>=</mo><msup><mi>C</mi><mi>T</mi></msup><mi>C</mi></mrow><annotation encoding="application/x-tex">A=C^TEC=C^TC</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8413309999999999em;"></span><span class="strut bottom" style="height:0.8413309999999999em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">A</span><span class="mrel">=</span><span class="mord"><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">T</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.05764em;">E</span><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="mrel">=</span><span class="mord"><span class="mord mathit" style="margin-right:0.07153em;">C</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">T</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.07153em;">C</span></span></span></span></p></li> <li><p>若A正定，则A的秩正定，则 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>A</mi><mi>T</mi></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">A^TA</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8413309999999999em;"></span><span class="strut bottom" style="height:0.8413309999999999em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">A</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">T</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit">A</span></span></span></span>正定</p></li> <li><p>只要实对称矩阵有相同的特征值，则他们相似且合同</p></li> <li><p>涉及到高阶可导且证明极限则考虑使用带拉格朗日余项的泰勒公式</p></li> <li><p>反常积分瑕点，看p小于1；无穷看p大于1</p></li> <li><p>泰勒公式： <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>+</mo><msub><mi>x</mi><mn>0</mn></msub><mo>)</mo><mo>=</mo><mo>∑</mo><mfrac><mrow><msup><mi>f</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><msubsup><mi>x</mi><mn>0</mn><mi>n</mi></msubsup></mrow><mrow><mi>n</mi><mo>!</mo></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>!</mo></mrow></mfrac><msup><mi>f</mi><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>(</mo><mi>ξ</mi><mo>)</mo><msubsup><mi>x</mi><mn>0</mn><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">f(x+x_0) = \sum \frac{f^{(n)} (x) x_0^n}{n!} + \frac{1}{(n+1)!}f^{(n+1)} (\xi)x_0^{n+1}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:1.11922em;"></span><span class="strut bottom" style="height:1.6392200000000001em;vertical-align:-0.52em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mbin">+</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span><span class="mrel">=</span><span class="op-symbol small-op mop" style="top:-0.0000050000000000050004em;">∑</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">n</span><span class="mclose">!</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.49012em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:-0.363em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord scriptscriptstyle uncramped"><span class="mopen">(</span><span class="mord mathit">n</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mclose">)</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.2573142857142857em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle cramped"><span class="mord mathrm">0</span></span></span><span style="top:-0.363em;margin-right:0.07142857142857144em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-scriptstyle scriptscriptstyle uncramped"><span class="mord mathit">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mbin">+</span><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.34500000000000003em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mopen">(</span><span class="mord mathit">n</span><span class="mbin">+</span><span class="mord mathrm">1</span><span class="mclose">)</span><span class="mclose">!</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mopen">(</span><span class="mord mathit">n</span><span class="mbin">+</span><span class="mord mathrm">1</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.04601em;">ξ</span><span class="mclose">)</span><span class="mord"><span class="mord mathit">x</span><span class="vlist"><span style="top:0.266308em;margin-left:0em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathrm">0</span></span></span><span style="top:-0.403131em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">n</span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></p></li> <li><p>第一句话：在题设条件中给出一个函数f(x)二阶或二阶以上可导时，“不管三七二十一”把f(x)在指定点展成泰勒公式再说</p></li> <li><p>第二句话：在题设条件或欲证结论中有定积分表达式时，则“不管三七二十一”先用积分中值定理对该积分式处理一下再说，</p></li> <li><p>第三句话：在题设条件中函数f(x) 在[a,b]上连续，在(a,b)内可导，且f(a)=0或f(b)=0，则“不管三七二十一”先用拉格朗日中值定理处理一下再说，考虑此形式：f(x) - f(a) = f’(\xi) (x - a)</p></li> <li><p>如果不等式证明出现2的幂，很可能推理过程用到取中点 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{a+b}{2}</annotation></semantics></math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8801079999999999em;"></span><span class="strut bottom" style="height:1.2251079999999999em;vertical-align:-0.345em;"></span><span class="base textstyle uncramped"><span class="mord reset-textstyle textstyle uncramped"><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathrm">2</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.394em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped"><span class="mord scriptstyle uncramped"><span class="mord mathit">a</span><span class="mbin">+</span><span class="mord mathit">b</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span></p></li> <li><p>第四句话：对定限或变限积分，若被积函数或其主要部分为复合函数，则“不管三七二十一先做变量替换使之成为简单形式f(u）再说，</p></li> <li><p>第五句话：在证明文字不等式时，“不管三七二十一”，将出现两次或两次以上的某个文字或数字改写成变量x，转化成函数不等式再说．</p></li> <li><p>求定积分，对于指数函数乘以幂函数，如果幂的次数小于0且不为-1，分部积分考虑对幂函数积分</p></li></ol></div> <footer class="page-edit"><!----> <div class="last-updated"><span class="prefix">Last Updated:</span> <span class="time">2021/2/11 下午10:55:23</span></div> <a rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.zh"><img alt="知识共享许可协议" src="" style="border-width:0"></a><br>本作品采用<a rel="license" href="http://creativecommons.org/licenses/by/4.0/">知识共享署名 4.0 国际许可协议</a>进行许可。

   
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